Linear Codes from Projective Varieties: A Survey

Abstract

Linear codes can be constructed from classical algebraic varieties or from appropriate subsets of finite geometry by considering projective systems arising from their rational points. This geometric point of view allows to look for linear codes by choosing suitable sets to get immediately length, minimum distance, and spectrum (cf. Lemma 1, Propositions 5, 9, 12, 13, 17). In some cases, it is also possible to build a PD-set or an antiblocking decoding (cf. Propositions 3, 4, 14, Examples of Section 5).

1. Introduction

In the construction of a code, it would be desirable to have a small length, great dimension, and minimum distance to keep the transmission rate low and to get both much information and the correction of many errors. But it is impossible to improve all the basic parameters at the same time.

The purpose of this article is to collect results on linear codes arising from classical varieties via projective systems, by adopting a geometric point of view. If the basic parameters of such varieties (or of appropriate subsets of points) in a projective space can be calculated directly, then it is possible to know immediately length and minimum distance and sometimes also the weight distribution of the related linear codes.

When the group of automorphisms of a variety is read in the automorphism group of a related code, in some cases, it is possible to construct a PD-set (Permutation Decoding set) and/or an AI-system (Antiblocking system).

A PD-set for a t-error-correcting code C is a set S of automorphisms of the code such that every possible error vector of weight $w\le t$ can be moved out of the information positions by some member of S (cf. [1, 2]). To apply a PD-set to decode a message refer to Huffman ([3], pp. 1345–1440), where an algorithm is given. The permutation decoding algorithm is more efficient the smaller size of the PD-set. The Gordon (lower) bound on this size is crucial (cf. [2] and [3], p. 1414).

A large automorphism group of a code allows to find a PD-set. Indeed, linear codes defined by projective systems usually have large automorphism group. In [4] is generalized the notion of a PD-set of a code to that of a t-PD-set of an arbitrary permutation set.

An AI-system (Antiblocking system) is a new decoding algorithm developed by Kroll and Vincenti in [5, 6], which is comparable to the permutation decoding algorithm, but more efficient being simpler and faster than the permutation decoding algorithm. The existence of a PD-set implies usually also the existence of an AI-system of the same size. But there also may exist AI-systems that are not derived from PD-sets and which are smaller than the known PD-sets. By comparing the two decoding algorithms, it is clear that the antiblocking decoding needs less computing steps than the permutation decoding, even if the size of both systems is the same. Moreover, the antiblocking decoding may be applied even if there does not exist a PD-set or if there exists only a PD-set of very large size and an AI-system of smaller size. For the technique to find small AI-systems, refer to [5], where some properties of antiblocking systems are established.

Codes related to quadrics in the 3-dimensional finite projective spaces PG(3, q) mark the way for the next examples. The geometries of the plane sections of the quadrics over a conic are well known, as well as their automorphism groups. In Section 3, PD-sets for $q=3$ for all three cases are presented. For the elliptic and hyperbolic quadric, also examples for $q=4$ are given. These results say that the corresponding codes admit PD-sets S. The size $\mid S\mid$ is minimal in some cases. For the hyperbolic quadric also a 5-AI-system is shown, while the Gordon bound is 6 (cf. Propositions 3, 4).

In Section 4, we will refer to the construction of linear codes arising, respectively, from the Grassmannian of the lines of the third dimension (that is, the Klein quadric) and from the Schubert variety of $\mathit{PG}\left(5,q\right)$ (cf. Proposition 5, 6 and Examples 1, 2).

In Section 5, we consider codes related to what we call the celtic variety, that is, the ruled rational normal surface $V_2^3$ of order three in $\mathit{PG}\left(4,q\right)$ (cf. Propositions 13). Examples of PD-sets for q = 3 and q = 4 are given in Proposition 14.

In the last Section, results concerning projective systems and codes related to ruled sets are collected (cf. Proposition 17 and Examples 3, 4, 5). In the examples, the weight distribution of each code is also shown.

The title of each section refers only to the varieties of which the related codes are described there.

2. Codes and projective systems

Let $F=\mathit{GF}\left(q\right)$ be a finite field, $q=p^s$, p prime, denote by $F^n$ the n-dimensional vector space over F.

A linear${\left[n,k\right]}_q$-code C of length n is a k-dimensional subspace of the vector space $F^n$.

For $t\ge 1$ the t-th higher weight of C (cf. Wei [7]) is defined by

$$d_t=d_t\left(C\right)=min\left\{\parallel D\parallel \text{for}\mathrm{all}D<C,dimD=t\right\},$$E1

where D is a subspace of C and $\parallel D\parallel$ is the number of indices i such that there exists $v\in D$ with $v_i\ne 0$.

The first parameter $d_1=d_1\left(C\right)$ is the Hamming distance d, that is, the classical minimum distance (or, minimum weight) of C.

The code C has genus at most $g\ge 0$ if $k+d_1\ge n+1-g$.

Sometimes an ${\left[n,k\right]}_q$-code C of minimum distance d is denoted ${\left[n,k,d\right]}_q$-code.

Let $P^{k-1}={\mathit{PrF}}^k=\mathit{PG}\left(q-1,q\right)$ denote the $\left(k-1\right)$-dimensional Galois projective space over the field F, $k\ge 3$ with point set $\mathcal{P}$ and line set $\mathfrak{L}$. Denote $\mathfrak{T}$ the set of all subspaces of $\mathit{PG}\left(k-1,q\right),\mathrm{ℌ}$ the set of the hyperplanes of $\mathit{PG}\left(k-1,q\right)$.

The incidence hull of a subset $X\subset \mathcal{P}$ is denoted by $\overline{X}$. Thus the joining line of two points $X,Y\in \mathcal{P}$ is $\overline{X,Y}≔\overline{\left\{X,Y\right\}}$.

An [n, k]-projective system$\mathcal{X}$ of $P^{k-1}$ is a collection of n points. $\mathcal{X}$ is non-degenerate if its n points are not contained in any hyperplane.

From now on assume that $\mathcal{X}$ consists of n distinct points of rank k.

A standard matrix$\mathcal{M}$ can be constructed as follows: for each of the n points of $\mathcal{X}$ choose a generating vector, such n vectors are the rows of $\mathcal{M}$. Let $C_{\mathcal{X}}$ be the linear code having ${\mathcal{M}}^t$ as a generatrix matrix. The code $C_{\mathcal{X}}$is the k-dimensional subspace of$F^n$ which is the image of the mapping${\left(F^k\right)}^{\ast }\twoheadrightarrow F^n$from the dual k-dimensional space${\left(F^k\right)}^{\ast }$onto$F^n$that calculates every linear form over the points of$\mathcal{X}$. Therefore the length n of a codeword $C_{\mathcal{X}}$ is the cardinality of $\mathcal{X}$, and the dimension of $C_{\mathcal{X}}$ is k.

An automorphism of the code C is a weight-preserving linear automorphism (cf. [4], Section 2).

The equivalences among ${\left[n,k,d\right]}_q$-codes are the restrictions of the automorphisms of $F^n$ represented by monomial matrices, where a monomial matrix is the product of a permutation matrix and a diagonal matrix (for the basic concepts of coding theory see for example [3]). To any subset representing the n points of $\mathcal{X}$ is associated a linear [n, k, d]-code, any two such ${\left[n,k,d\right]}_q$-linear codes are equivalent.

A natural 1–1 correspondence connects the equivalence classes of a non-degenerate ${\left[n,k\right]}_q$-projective system$\mathcal{X}$ with a non-degenerate ${\left[n,k\right]}_q$-code$C_{\mathcal{X}}$. If $\mathcal{X}$ is an ${\left[n,k\right]}_q$-projective system and $C_{\mathcal{X}}$ is a corresponding code, then the non-zero codewords of $C_{\mathcal{X}}$ correspond to hyperplanes $\mathrm{ℌ}$ of $P^{k-1}$, up to a non-zero factor, the correspondence preserving the parameters $n,k,d_t$. Therefore, the weight of a codeword c corresponding to the hyperplane $H_{\mathbf{c}}$ is the number of points of $\mathcal{X}\backslash H_{\mathbf{c}}$ so that the minimum weight d of the code $C_{\mathcal{X}}$ is $d=\mid \mathcal{X}\mid -\mathrm{m}\mathit{ax}\left\{|\mathcal{X}\cap H|||H\in \mathrm{ℌ}\right\}$.

A linear code C having d as minimum weight is an s-error-correcting code for all $s\le ⌊\frac{d-1}{2}⌋$, and $t=⌊\frac{d-1}{2}⌋$ is the error-correcting capability of C.

Subcodes D of C of dimension r correspond to subspaces of codimension r of $P^{k-1}$. Therefore the higher weights of C are $d_t=d_t\left(C\right)=n-max\left\{||\mathcal{X}\cap S|:S<P^{k-1},\text{codim}S=t\right\}$. In particular, $d_1=d_1\left(C\right)=n-max\left\{||\mathcal{X}\cap H|:H<P^{k-1},\text{codim}H=1\right\}$.

The spectrum of a projective system $\mathcal{X}$ of $P^{k-1}$ is the set of the following numbers $A_i^{\left(s\right)}=\mid \left\{S<P^{k-1}:\text{codim}S=s,||S\cap \mathcal{X}||=n-i\right\}\mid$ for all $i=1,2,\dots ,n,s=1,2,\dots ,k-2$.

Let $H\in \mathrm{ℌ}$ be a hyperplane. An intersection number of$\mathcal{X}$(with respect to hyperplanes) is $\mid \mathcal{X}\cap H\mid$. The type of$\mathcal{X}$with respect to the hyperplanes is the set $M_{\mathcal{P}}$ of all intersection numbers of $\mathcal{X}$. For $i\in M_{\mathcal{P}},t_i≔\left\{H\in \mathrm{ℌ}|||\mathcal{X}\cap H|=i\right\}$ is the total number of hyperplanes providing the intersection number i.

Let $\mathcal{X}$ be a projective system of type $M_{\mathcal{P}}$. Then for $i\in M_{\mathcal{P}}$there are$t_i$code words in the related code$C_{\mathcal{X}}$of weight$\mid \mathcal{X}\mid -i$. Analogous definitions can be stated for all subspaces of $\mathfrak{T}$. Therefore, the spectrum of $\mathcal{X}$ induces the weight distribution of the codewords of$C_{\mathcal{X}}$.

For the above definitions see also [8, 9, 10].

For the definitions of the permutation and the antiblocking decoding and the respective algorithms, see [5, 6], p. 1463.

The following result holds (cf. [8]):

Result A (non-degenerate) projective system of $P^k$ satisfies the following

Gordon bound: (cf. [2, 11]) Let S be a PD-set for a t-error-correcting [n, k]-code with redundancy $r=n-k$. Then

$$\mid S\mid \ge ⌈\frac{n}{r}⌈\frac{n-1}{r-1}\dots ⌈\frac{n-t+1}{r-t+1}⌉\dots ⌉⌉.$$E2

Following the geometric interpretation of a linear code shown by Tsfasman and Vladut (cf. [12, 13]) many authors studied codes arising from sets of the rational points of an affine or of a projective space.

Denote $\overline{F}$ the algebraic closure of the field $F=\mathit{GF}\left(q\right)$.

The geometry $\mathit{PG}\left(r,q\right)=P^r$ is considered a sub-geometry of $\overline{\mathit{PG}}\left(r,q\right)={\overline{P}}^r$, projective geometry over $\overline{F}$. We refer to the points of $P^r$ as the rational points of ${\overline{P}}^r$.

A variety$V_u^v$of dimension u and of order v of $P^r$ is the set of the rational points of a projective variety ${\overline{V}}_u^v$ of ${\overline{P}}^r$ defined by a finite set of polynomials of $F\left[x_0,\dots ,x_r\right]$.

Choose a coordinate system in $P^r$ so that it is a coordinate system for ${\overline{P}}^r$ too, denote a point $P\approx \left(x_0,x_1,\dots ,x_r\right)≔{\overline{F}}^{\ast }\left(x_0,x_1,\dots ,x_r\right),{\overline{F}}^{\ast }=\overline{F}\backslash \left\{0\right\}$.

P is a rational point if there exists $\left(x_0,x_1,\dots ,x_r\right)\in F^{r+1}$ such that $P\approx \left(x_0,x_1,\dots ,x_r\right)$.

For the definition of projective variety, the reader can refer to [14, 16].

If $\mathcal{X}$ is a subset of $n=\mid \mathcal{X}\mid$ points of $P^r$, then a subspace of $P^r$ of projective dimension u is denoted by $S_u$. A variety of dimension u and of order v is denoted $V_u^v$.

A t-secant subspace is a subspace intersecting $\mathcal{X}$ in t points. A t-secant is a t-secant line.

A line t is a tangent of $\mathcal{X}$ if either t has just one point in common with $\mathcal{X}$ or, each point of t is contained in $\mathcal{X}$. If a tangent line t has just one point in common with $\mathcal{X}$, then t is a tangent of$\mathcal{X}$at the point P. A subspace $\mathcal{U}$ is a component-subspace if each point of $\mathcal{U}$ lies in $\mathcal{X}$. A line l with the property that each of its points lies in $\mathcal{X}$ is a component-line, or simply a component. If $m-1$ is the largest dimension of a component space of $\mathcal{X}$, then m is the vector index of $\mathcal{X}$.

3. The quadrics in PG$\left(3,q\right)$

In the 3-dimensional geometry $P^3=\mathit{PG}\left(3,q\right)$ over the Galois field $F=\mathit{GF}\left(q\right)$ we will consider the elliptic quadric${\mathcal{E}}_3$, the hyperbolic quadric${\mathcal{H}}_3$, and the quadric cone${\mathcal{Q}}_{\mathcal{O}}$ with vertex $\mathcal{O}$ and directrix a conic of a plane $\pi ,\mathcal{O}\notin \pi .$.

Denote $\mathcal{Q}$ the projective system defined by the rational points of a quadric $\mathit{Q}$. Let ${\mathcal{C}}_Q$ be a related code.

The minimum weights $d_t$ of the projective system $\mathcal{Q}$ consisting of n points are, in such a dimension, $d_t=n-max\left\{|\mathcal{Q}\cap S_t|\right\},t=1,2$,

and the spectrum of $\mathcal{Q}$ is

$$A_{n-i}^{\left(s\right)}=\mid \left\{S_{3-s}<\mathit{PG}\left(3,q\right):|S_{3-s}\cap \mathcal{Q}|=i\right\}\mid$$

$$i=1,2,\dots ,n,s=1,2.$$E3

To construct a linear code related to a quadric, one needs to know all the intersection numbers related to it.

From [14] we get

1. the elliptic quadric ${\mathcal{E}}_3:f\left(x_0,x_1\right)+x_2{x}_3=0,$ consists of $q^2+1$ points no three of which are collinear, $q^2+1$ tangent planes, $q\left(q^2+1\right)$ and secant planes;

2. the hyperbolic quadric ${\mathcal{H}}_3:x_0{x}_1+x_2{x}_3=0$ consists of ${\left(q+1\right)}^2=q^2+2q+1$ points on $2\left(q+1\right)$ (component) lines, each point on two lines, $q\left(q^2-1\right)$ secant planes through conics;

3. the cone ${\mathcal{Q}}_{\mathcal{O}}:x_0^2+x_1{x}_4=0,$ consists of $q\left(q+1\right)+1=q^2+q+1$ points on the $q+1$ (component) lines projecting from the point $\mathcal{O}$ a conic directrix.

Let H be a plane of $\mathrm{ℌ}$. If $\mathcal{Q}$ is elliptic, then $\mathcal{Q}\cap H$ is a point or a conic. If $\mathcal{Q}$ is hyperbolic, then $\mathcal{Q}\cap H$ is the union of two lines or a conic. If $\mathcal{Q}$ is a cone, then $\mathcal{Q}\cap H$ is the vertex $\mathcal{O}$ or a line or the union of two lines or a conic.

From [14] and from above the basic parameters and the spectrum of the projective system $\mathcal{Q}$ can be shown.

Lemma 1 (1) If$\mathcal{Q}={\mathcal{E}}_3$then$n=q^2+1,k=4,d_1=q^2-q,d_2=q^2-1;$

$A_{n-\left(q+1\right)}^{\left(1\right)}=q\left(q^2+1\right),A_{n-1}^{\left(1\right)}=q^2+1,A_{n-2}^{\left(2\right)}=q^2\frac{\left(q^2+1\right)}{2},A_{n-1}^{\left(2\right)}=\left(q+1\right)\left(q^2+1\right),A_n^{\left(2\right)}=q^2\frac{\left(q^2+1\right)}{2}$and no other parameter is different from zero.

(2) If$\mathcal{Q}={\mathcal{H}}_3$then$n={\left(q+1\right)}^2,k=4,d_1=q^2,d_2=q^2+q;$

$$A_{n-\left(2q+1\right)}^{\left(1\right)}={\left(q+1\right)}^2,A_{n-\left(q+1\right)}^{\left(1\right)}=q\left(q^2-1\right);$$E4

$A_{n-\left(q+1\right)}^{\left(2\right)}=2\left(q+1\right),A_{n-2}^{\left(2\right)}=q^2\frac{{\left(q+1\right)}^2}{2},A_{n-1}^{\left(2\right)}=\left(q+1\right)\left(q^2-1\right),A_n^{\left(2\right)}=\frac{q{\left(q-1\right)}^2}{2}$and no other parameter is different from zero.

(3) If$\mathcal{Q}={\mathcal{Q}}_{\mathcal{O}}$then$n=q^2+q+1,k=4,d_1=q^2-q,d_2=q^2;$

$$A_{n-\left(2q+1\right)}^{\left(1\right)}=\frac{q\left(q+1\right)}{2},A_{n-\left(q+1\right)}^{\left(1\right)}=q^3+q+1,A_{n-1}^{\left(1\right)}=\frac{q\left(q-1\right)}{2};$$E5

$$A_{n-\left(q+1\right)}^{\left(2\right)}=q+1,A_{n-2}^{\left(2\right)}=\frac{q^3\left(q+1\right)}{2},$$E6

$A_{n-1}^{\left(2\right)}=q^3+q^2,A_n^{\left(2\right)}=\frac{q^3\left(q-1\right)}{2}$and no other parameter is different from zero.

Then is proved the following (cf. Proposition 5 of [4]).

Proposition 2 (1) If$\mathcal{Q}={\mathcal{E}}_3$, then$\mathcal{Q}$is a${\left[q^2+1,4,q\left(q-1\right)\right]}_q$-projective system. 2) If$\mathcal{Q}={\mathcal{H}}_3$, then$\mathcal{Q}$is a${\left[{\left(q+1\right)}^2,4,q^2\right]}_q‐\mathit{\text{projective}}$system. 3) If $\mathcal{Q}={\mathcal{Q}}_{\mathcal{O}},$then$\mathcal{Q}$is a${\left[q\left(q+1\right),4,q\left(q-1\right)\right]}_q$-projective system.

The following propositions supply examples for $q=3$ and $q=4$. Denote $\mathcal{Q}$ a quadric in $\mathit{PG}\left(3,3\right)$ and ${\mathcal{C}}_Q$ a related code. From [4], Proposition 11 we get

Proposition 3 (1) If$\mathcal{Q}={\mathcal{E}}_3$, then${\mathcal{C}}_Q$is a 2-error-correcting${\left[\mathrm{10,4,6}\right]}_3$-code admitting a PD-set S of minimum size 4.

(2) If$\mathcal{Q}={\mathcal{H}}_3$, then${\mathcal{C}}_Q$is a 4-error-correcting${\left[\mathrm{16,4,9}\right]}_3$-code admitting a PD-set of size 8.

(3) If${\mathcal{Q}}^{\prime }={\mathcal{Q}}_{\mathcal{O}}\backslash \left\{\mathcal{O}\right\}$, then a related code${\mathcal{C}}_Q^{\prime }$is a 2-error-correcting${\left[\mathrm{12,4,6}\right]}_3$-code admitting a PD-set S of minimum size 3.

Denote $\mathcal{Q}$ a quadric in $\mathit{PG}\left(3,4\right)$ and ${\mathcal{C}}_Q$ a related code. From [4], Propositions 12, 13 and from the Example of [6], p.1464, we get

Proposition 4 (a) If$\mathcal{Q}$is an elliptic quadric, then${\mathcal{C}}_Q$is a 5-error-correcting-${\left[\mathrm{17,4,12}\right]}_4$-code admitting a PD-set$\mathcal{S}$of size 16.

Let $\mathcal{Q}$ be a hyperbolic quadric. Then

$\left(b_1\right)$${\mathcal{C}}_Q$is a${\left[\mathrm{25,4,16}\right]}_4$-code admitting a PD-set$\mathcal{S}$of size 20 and a 5-AI-system$\mathcal{A}$.

$\left(b_2\right)$If$P\in \mathcal{Q}$and [P] is the union of the two generators of$\mathcal{Q}$passing through P, then${\mathcal{Q}}^{\prime }=\mathcal{Q}\backslash \left[P\right]$gives rise to a code${\mathcal{C}}_{Q^{\prime }}$being a${\left[\mathrm{16,4,9}\right]}_4$-code admitting a PD-set$\mathcal{S}$of size 12.

Note that in case $\left(b_1\right)$, the Gordon bound is 6.

4. The Klein quadric and the Schubert variety of $\mathit{PG}\left(5,q\right)$

Let ${\mathfrak{U}}_l$ be the set of all l-dimensional subspaces of $\mathit{PG}\left(r,q\right)$. The Grassmann mapping$\mathfrak{G}:{\mathfrak{U}}_l\to \mathit{PG}\left(N,q\right),N=\left(\begin{array}{l}r+1\\ l+1\end{array}\right)-1,$ associates to any $\mathcal{U}\in {\mathfrak{U}}_l$ a point $\mathfrak{G}\left(\mathcal{U}\right)$ of $\mathit{PG}\left(N,q\right)$. Then $\mathit{im}\mathfrak{G}={\mathcal{G}}_{l,r}≔\mathfrak{G}\left({\mathfrak{U}}_l\right)$ is an algebraic variety called the Grassmannian of the l-dimensional subspaces of $\mathit{PG}\left(r,q\right)$ (cf. [16], p. 107). The number of points of ${\mathcal{G}}_{l,r}$ is given by

$$\mid {\mathcal{G}}_{l,r}\mid =\left[\begin{array}{c}r+1\\ l+1\end{array}\right]=\frac{\left(q^{r+1}-1\right)\left(q^r-1\right)\dots \left(q^{r-l+1}-1\right)}{\left(q^{l+1}-1\right)\left(q^l-1\right)\dots \left(q-1\right)}.$$E7

The Grassmannian ${\mathcal{G}}_{1,3}$ of the lines of $\mathit{PG}\left(3,q\right)$ is the hyperbolic quadric $\mathcal{K}Q$ of $\mathit{PG}\left(5,q\right)$ consisting of $\left(q^2+1\right)\left(q^2+q+1\right)$ points. It is called the Klein quadric.

It has a projective index 2, and it is covered by two systems of component planes. For general details see [14, 15, 16], for details on the intersection properties see [17] Section 4).

A linear code related to $\mathcal{KQ}$ is an [n, k]-code where $n=\left(q^2+1\right)\left(q^2+q+1\right),k=6$ and minimum distance $d=q^4$ (see [18], p. 147, [13], p. 1579]).

Let $\mathcal{X}={\mathcal{X}}_{\mathcal{KQ}}$ denote the projective system associated to $\mathcal{KQ}$. As usual, the basic parameters and spectrum are denoted respectively n, k, $d_t=d_t\left(\mathcal{X}\right)$ and $A_{n-i}^{\left(s\right)}=A_{n-i}^{\left(s\right)}\left(\mathcal{X}\right)$ with $s,t=\mathrm{1,2,3,4}$.

By direct computation we get

Proposition 5.$\mathcal{X}$has the following basic parameters and spectrum:

(1) $n=\left(q^2+1\right)\left(q^2+q+1\right),k=6,$

$$d_1=q^4,d_2=q^4+q^3,d_3=q^4+q^3+q^2,d_4=q^4+q^3+2q^2;$$E8

(2) with respect to the hyperplanes:

$A_{n-\left(q^3+2q^2+q+1\right)}^{\left(1\right)}=q^4+q^3+2q^2+q+1$(hyperplanes cutting Schubert varieties);

$A_{n-\left(q^3+q^2+q+1\right)}^{\left(1\right)}=q^5-q^2$(hyperplanes cutting parabolic quadrics of the 4th dimension);

and $A_{n-j}^{\left(1\right)}=0,j\ne q^3+q^2+q+1\mathit{\text{or}}j\ne q^3+2q^2+q+1.$

(3) with respect to the solids:

$A_{n-\left(2q^2+q+1\right)}^{\left(2\right)}=\left(q^3+q^2+q+1\right)\left(q^2+q+1\right)$(solids cutting pairs of component planes);

$A_{n-{\left(q+1\right)}^2}^{\left(2\right)}=\frac{1}{2}{q}^4\left(q^4+q^3+2q^2+q+1\right)$(solids cutting hyperbolic quadrics of the 3th dimension);

$A_{n-\left(q^2+q+1\right)}^{\left(2\right)}=\left(q^3-q\right)\left(q^2+1\right)\left(q^2+q+1\right)$(solids cutting cones of the 3th dimension);

$A_{n-\left(q^2+1\right)}^{\left(2\right)}=\frac{1}{2}{q}^4\left(q^3-1\right)\left(q-1\right)$(solids cutting elliptic quadrics of the 3th dimension);

and $A_{n-j}^{\left(2\right)}=0,j\ne {\left(q+1\right)}^2,\left(q^2+1\right),\left(q^2+q+1\right),\left(2q^2+q+1\right).$

(4) with respect to the planes:

$A_{n-\left(q^2+q+1\right)}^{\left(3\right)}=2\left(q^3+q^2+q+1\right)$(component planes);

$A_{n-\left(2q+1\right)}^{\left(3\right)}=\frac{1}{2}{q}^2{\left(q+1\right)}^2\left(q^2+1\right)\left(q^2+q+1\right)$(planes cutting two component lines);

$A_{n-\left(q+1\right)}^{\left(3\right)}=\left(q^4\left(q^3-1\right)\left(q^2+1\right)\right)+\left(\left(q^4-1\right)\left(q^2+q+1\right)\right)=c+r$(planes cutting one conic (c conics) or one line (r lines));

$A_{\left(n-1\right)}^{\left(3\right)}=\frac{1}{2}{q}^2\left(\left(q^4+1\right)\left(q^2-q+1\right)-2q^3\right)$(planes cutting one point);

and$A_{\left(n-j\right)}^{\left(3\right)}=0,j\ne \left(q^2+q+1\right),\left(2q+1\right),\left(q+1\right),1$(there are no s-secant planes for$s\in \left\{2,3,\dots ,q\right\}).$.

(5) with respect to the lines:

$A_{n-\left(q+1\right)}^{\left(4\right)}=\left(q^2+1\right)\left(q^2+q+1\right)\left(q+1\right)$(component lines);

$A_{n-2}^{\left(4\right)}=\frac{1}{2}{q}^4\left(q^4+q^3+2q^2+q+1\right)$(2-secant lines);

$A_{n-1}^{\left(4\right)}=\left(q^3-q\right)\left(q^2+1\right)\left(q^2+q+1\right)$(tangent lines);

$A_{n-0}^{\left(4\right)}=\frac{1}{2}{q}^4\left(q^3-1\right)\left(q-1\right)$(external lines);

and$A_{n-j}^{\left(4\right)}=0,j\ne \left(q+1\right),2,1,0$.

From Section 2, it is clear that the previous Proposition 5 provides the complete spectrum of a linear code ${\mathcal{C}}_{\mathcal{X}}$ related to the Klein quadric.

In Section 5 of [19] is shown the following example.

Example 1 A binary linear code ${\mathcal{C}}_{\mathcal{KQ}}$ related to the Klein quadric $\mathcal{KQ}$ in $\mathit{PG}\left(5,2\right)$ is a [35, 6]-code with minimum distance $d=2^4=16$ and admits a PD-set of size 40.

A Schubert variety${\mathcal{S}}_{\mathcal{KQ}}$ of $\mathit{PG}\left(5,q\right)$ is a section of the Klein quadric $\mathcal{KQ}$ by a tangent hyperplane T. Thus ${\mathcal{S}}_{\mathcal{KQ}}$ is a cone of $2\left(q+1\right)$ planes with vertex $O\in \mathcal{KQ}$ and consists of $n=q^3+2q^2+q+1$ points. It corresponds via the Grassmann mapping $\mathfrak{G}$ to a special linear complex of lines of $\mathit{PG}\left(3,q\right)$, that is, a set comprising all lines meeting a fixed line. It is unique up to projectivities (cf. [14, 15, 16]).

If $\left(x_0,\dots ,x_5\right)$ are projective coordinates in $\mathit{PG}\left(5,q\right)$, the quadric $\mathcal{KQ}$ can be represented by the equation $x_0{x}_5-x_1{x}_4+x_2{x}_3=0$ so that a Schubert variety ${\mathcal{S}}_{\mathcal{KQ}}$ is the section of $\mathcal{KQ}$ by the hyperplane T with the equation $x_5=0$. Then ${\mathcal{S}}_{\mathcal{KQ}}=\mathcal{KQ}\cap T$ is a cone section with vertex the point (1, 0, 0, 0, 0) from which the hyperbolic quadric ${\mathcal{H}}_3:x_1{x}_4-x_2{x}_3=0$ of a solid is projected.

To calculate $d_1$ it is easy to verify that the maximum intersection with hyperplanes H of $\mathrm{ℌ}$ is obtained when H contains one of the planes and meets each of the remaining q planes in lines, all passing through the vertex. Hence the maximum intersection is $2q^2+q+1$.

To calculate $d_2$ the maximum intersection with planes (2-dimensional subspaces of $\mathfrak{T}$) is obtained if a plane is one of the ruling planes or is a plane through the vertex which meets every ruling plane in a line. In any case, we get $q^2+q+1$.

Finally $d_3$ is clear as in ${\mathcal{S}}_{\mathcal{KQ}}$ there are component lines.

From above is proved the following result.

Proposition 6. The basic parameters and weights of${\mathcal{S}}_{\mathcal{KQ}}$are

$$n=q^3+2q^2+q+1,k=5,d_1=q^3=n-2q^2+q+1,$$E9

$$d_2=n-q^2+q+1=q^3+q^2,d_3=n-\left(q+1\right)=q^3+2q^2.$$E10

If H is a hyperplane of $\mathrm{ℌ}$, then $\mid {\mathcal{S}}_{\mathcal{KQ}}\cap H\mid ={\left(q+1\right)}^2,$ or $\mid {\mathcal{S}}_{\mathcal{KQ}}\cap H\mid =2q^2+q+1$ according to $O\notin H$ or $O\in H$, respectively. Therefore linear codes related to ${\mathcal{S}}_{\mathcal{KQ}}$ and to $\mathcal{V}={\mathcal{S}}_{\mathcal{KQ}}\backslash \left\{O\right\}$ (both of dimension 5) have the same minimum distance $d=q^3$. Since the automorphisms group $\mathrm{Aut}{\mathcal{S}}_{\mathcal{KQ}}$ fixes the vertex O, a code related to$\mathcal{V}$has better parameters than a code related to${\mathcal{S}}_{\mathcal{KQ}}$.

In Section 5 of [19] is shown the following example.

Example 2. A binary linear code ${\mathcal{C}}_{\mathcal{V}}$ related to the Schubert variety $\mathcal{V}={\mathcal{S}}_{\mathcal{KQ}}\backslash \left\{O\right\}$ in $\mathit{PG}\left(5,2\right)$ is a [18, 5]-code with minimum distance $d=2^3=8$ and admits a PD-set of size 9.

To get some information about the Grassmannian ${\mathcal{G}}_{l,r}$ of the l-dimensional subspaces in $\mathit{PG}\left(r,q\right),r>5,$ and the Schubert variety $\mathrm{\Omega }\left(\alpha \right)\subset {\mathcal{G}}_{l,r}$ (where $\alpha =\left(a_0,\dots ,a_l\right)$ is the corresponding sequence of dimensions) and their codes, see Theorem 12 of [9, 20, 21, 22].

5. The rational ruled surfaces $V_2^{r-1}$ of PG$\left(r,q\right)$

Let us consider varieties of $P^r$ with $u=2$ and $v=r-1$.

The following result is well known (see [15]).

Proposition 7The varieties$V_2^{r-1}$of$P^r$are the rational ruled varieties and the Veronese surface if$r=5$.

Suitably modified it can be easily proved also for the finite case.

Assume $r\ne 5$. Denote $S_t$ a projective t-dimensional subspace of $P^r$ for $t<r$.

Proposition 8

1. $V_2^{r-1}$is a ruled rational normal surface.

2. A variety$V_2^{r-1}$contains irreducible rational normal curves$C^t$of order$t\le r-1$each of them existing in a t-dimensional subspace$S_t$.

3. Each$V_2^{r-1}$is a ruled surface obtained by means of a projectivity between two irreducible directrix curves$C^m$and$C^{r-m-1}$contained in anm-dimensional subspace$S_m$and in an$\left(r-m-1\right)$-dimensional subspace$S_{r-m-1}$, respectively.

4. Let m be the order of the minimum order directrix of$V_2^{r-1}$. Thenhgeneratrix lines are dependent if$h\le m+1$, otherwise they are independent.

5. If$r=2s$, then$m=\left(r-2\right)/2$is the order of the minimum order directrix, if$r=2s+1$, then there exist directrix curves of order$m\le \left(r-1\right)/2$.

Choose and fix a surface $V_2^{r-1}$ with a minimum order directrix $C^m$ with $m<q$.

Denote $\mathcal{X}$ the projective system of the rational points of $V_2^{r-1}$, C the linear code related to $\mathcal{X}$. It is $\mid \mathcal{X}\mid ={\left(q+1\right)}^2$.

Proposition 9C is an${\left[n,k,d\right]}_q$-code with

$$n={\left(q+1\right)}^2,k=r+1,d=d_1=q^2-\mathit{mq},d_{r-1}=q^2+q.$$E11

For the proof see [20], Theorem 4.

From $d_1\le n-k+1$, the definition of genus of a code and from Proposition 9 follows

Proposition 10 (1) The inequality$\left(m+2\right)q\ge r-1$holds for every q and r.

(2) C is of genus at most$g\ge \left(m+2\right)q-\left(r-1\right)$

Consider now the case $r=4$. Denote $\mathrm{ℌ}$ the set of the hyperplanes. Let $V_2^3$ be a ruled surface of $P^4=\mathit{PG}\left(4,q\right)$. We have named $V_2^3$ the celtic variety for its hut shape (see [4], Section 4).

From Proposition 8, from Lemma 7 of [20] and Propositions 1.1, 1.3, 1.4, 1.5, and Theorem 1.2 of [23] we obtain

Lemma 11

1. A celtic variety$V_2^3$is constructed by means of a projectivity connecting the points of the minimum order directrix, the line l, with the points of a non-degenerate conic$C^2$of a plane$\pi$with$\pi \cap l=\cancel{0}$.

2. $V_2^3$has$q+1$two by two skew generatrix lines. They connect birationally the points of l and the points of$C^2$.

3. There exists a unique hyperplane H such that$l\subset H$and$l^{\prime }=\pi \cap H$is skew to l.

4. There exist hyperplanes$H^{\prime }$such that one generatrix line$g_1$belongs to$H^{\prime }$and$H^{\prime }\cap V_2^3=\left\{g_1,{C^{\prime }}^2\right\}$for some conic${C^{\prime }}^2$.

5. For every two generatrix lines$g_i,g_j,i\ne j$the hyperplane$H^{\prime }=g_i\cup g_j$is such that$H^{\prime }\cap V_2^3=\left\{g_i,g_j,l\right\}$. Such hyperplanes have the maximum intersection with$V_2^3$.

6. No hyperplane contains 3 generatrix lines.

7. Every two points$P,Q\in V_2^3$belong to l, or to a generatrix line g, or to a unique conic of$V_2^3$.

8. A plane${\pi }^{\prime }$can meet$V_2^3$either in one point, or in one line, or in one irreducible conic, or it is the intersection of l with e generatrix line g. A plane${\pi }^{\prime }=l\cup g$is a tangent plane and$\mid {\pi }^{\prime }\cap V_2^3\mid$is maximum.

9. The totality of varieties$V_2^3$of$\mathit{PG}\left(4,q\right)$having l and$C^2$as directrices are projectively equivalent and their number is$\left(q+1\right)q\left(q-1\right)$.

Denote $\mathcal{X}$ the projective system consisting of the rational points of $V_2^3$ and $C_{\mathcal{X}}$ a linear code associated to it.

From Proposition 9, Proposition 10, 2), Lemma 11, (a), (h) and from [20], Theorems 8 and 9, we obtain

Proposition 12$C_{\mathcal{X}}$is an${\left[n,k\right]}_q$-code with$n={\left(q+1\right)}^2,k=5,d_1=q^2-q,d_2=q^2,d_3=q^2+q.C_{\mathcal{X}}$ (and$C_{\mathcal{X}}^{\perp })$is of positive genus$g\ge 3q-3$.

The spectrum$A_i^{\left(1\right)}$of$\mathcal{X}$is

$$A_{d_1}^{\left(1\right)}=\left(q+1\right)\frac{q}{2},A_{d_2}^{\left(1\right)}=\left(q^2-q\right)\left(q+1\right),A_{d_3}^{\left(1\right)}=\left(q^4+1\right)+\frac{q\left(q+3\right)}{2},$$E12

$$A_i^{\left(1\right)}=0\mathit{\text{for}}\mathit{all}i\in \left\{1,2,\dots ,n\right\}\backslash \left\{d_1,d_2,d_3\right\}.$$E13

Denote $g_s$ the generatrix line joining corresponding points $L_s\in l$ and $C_s\in C$. As l is the unique line intersecting all generatrices and there are no other lines contained in $\mathcal{X}$ than l and the generatrix, follows that every automorphism $\alpha \in \mathrm{Aut}\mathcal{X}$ fixes the directrix line l and maps every generatrix $g_s$ to a generatrix $g_s^{\prime }$ (cf. [4], Lemma 3).

From Lemma 11 follows that the intersection of $\mathcal{X}$ with a hyperplane H is the union of a generatrix and a conic, or the union of two generatrices and l, or the union of one generatrix and l, or l, or a cubic curve. Hence $max\left\{|\mathcal{X}\cap H||H\in \mathrm{ℌ}\right\}=3q+1$.

In order to construct PD-sets for the related codes, in Proposition 14 of [4], two subgroups of $\mathrm{Aut}\mathcal{X}$, namely $\mathcal{A}$ and $\mathcal{N}$, are chosen. $\mathcal{A}$ is isomorphic to the group of the affine bijections of F: $\left\{x|x\mapsto \mathit{xm}+b,m,b\in F,m\ne 0\right\},\mathcal{N}$ fixing each generatrix line is a normal subgroup.

Let ${\mathcal{X}}^{\prime }=\mathcal{X}\backslash \left\{l\right\}$. Note that $\mid {\mathcal{X}}^{\prime }\mid =q^2+q$ and $max\left\{|{\mathcal{X}}^{\prime }\cap H||H\in \mathrm{ℌ}\right\}=3q+1-\left(q+1\right)=2q$ so that the codes $C_{\mathcal{X}}$ and $C_{{\mathcal{X}}^{\prime }}$ have the same minimum distance.

As $\mathcal{X}$ generates $\mathit{PG}\left(4,q\right)$, choose a subset $\mathcal{I}\subset \mathcal{X}$ of independent points.

From above and by comparing [4], Proposition 15, we get the following result.

Proposition 13 (1) $C_{\mathcal{X}}$is a${\left[{\left(q+1\right)}^2,5,q\left(q-1\right)\right]}_q$-code.

(2) $C_{{\mathcal{X}}^{\prime }}$is a${\left[q\left(q+1\right),5,q\left(q-1\right)\right]}_q$-code.

(3) If$q\ge 4,\mathcal{I}\subset \mathcal{X}$an independent set of$\mathit{PG}\left(4,q\right)$with$\mathcal{I}\cap l\ne \cancel{0}$, then there is no PD-set for$\mathcal{I}$.

From [4], Propositions 17 and 19 follows

Proposition 14 (1) If$\mathcal{X}$is in$\mathit{PG}\left(4,3\right)$, then$C_{\mathcal{X}}$is a 2-error-correcting${\left[\mathrm{16,5,6}\right]}_3$-code admitting a PD-set S of minimum size 3.

(2) If$\mathcal{X}$is in$\mathit{PG}\left(4,4\right)$and${\mathcal{X}}^{\prime }=\mathcal{X}\backslash l$, then the code$C_{{\mathcal{X}}^{\prime }}$is a 5-error-correcting${\left[\mathrm{20,5,12}\right]}_4$-code admitting a PD-set S of size 24.

6. Ruled sets

Let $\mathit{PG}\left(k-1,q\right)=\left(\mathcal{P},\mathfrak{L}\right)$ be a $\left(k-1\right)$-dimensional projective space over $F=\mathit{GF}\left(q\right),k\ge 3$ with point set $\mathcal{P}$ and line set $\mathfrak{L}$. Denote $\mathrm{ℌ}$ the set of the hyperplanes of $\mathit{PG}\left(k-1,q\right)$.

Let $\mathcal{K}\subset \mathcal{P}$. Denote $M_{\mathcal{P}}$ the type of$\mathcal{K}$with respect to hyperplanes (that is, the set of all intersection numbers of $\mathcal{K}$). For $i\in M_{\mathcal{P}}$ let $t_i≔\left\{H\in \mathrm{ℌ}|||\mathcal{K}\cap H|=i\right\}$ denote the total number of hyperplanes yielding the intersection number i.

If $\mathcal{K}$ is a projective system of type $M_{\mathcal{P}}$, then for $i\in M_{\mathcal{P}}$ there are $t_i$ code-words in $C_{\mathcal{K}}$ of weight $\mid \mathcal{K}\mid -i$.

From Lemma 1 of [24] we obtain

Lemma 15Let$\mathcal{S}\subset \mathcal{P}$be a subspace with$\cancel{0}\ne \mathcal{S}\ne \mathcal{P}$and$\mathcal{K}\subset \mathcal{S}$. Then$M_{\mathcal{P}}=M_{\mathcal{S}}\cup \left\{|\mathcal{K}|\right\}$.

Let $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ be two complementary subspaces in $\mathit{PG}\left(k-1,q\right)$. Choose and fix two subsets $\mathcal{K}\subset \mathcal{S}$ and ${\mathcal{K}}^{\prime }\subset {\mathcal{S}}^{\prime }$ with $\overline{\mathcal{K}}=\mathcal{S},\overline{{\mathcal{K}}^{\prime }}={\mathcal{S}}^{\prime }.$ Set $m≔\mid \mathcal{K}\mid ,m^{\prime }≔\mid {\mathcal{K}}^{\prime }\mid$.

Denote $\mathcal{R}=\left\{\overline{x,x^{\prime }}|x\in \mathcal{K},x^{\prime }\in {\mathcal{K}}^{\prime }\right\}$. A ruled set$\mathcal{X}$ is the set of the points of the lines of $\mathcal{R}$, that is, $\mathcal{X}≔\bigcup _{X\in \mathcal{R}}X$.

From [24], Lemmas 2 and 3 follows

Lemma 16

1. Let$x_1,x_2\in \mathcal{K}$and$x_1^{\prime },x_2^{\prime }\in {\mathcal{K}}^{\prime }$with$x_1\ne x_2,x_1^{\prime }\ne x_2^{\prime }$; then$\overline{x_1,x_1^{\prime }}\cap \overline{x_2,x_2^{\prime }}=\cancel{0}.$

2. Let$L_1,L_2\in \mathcal{R},L_1\ne L_2$with$L_1\cap L_2\ne \cancel{0};$then$L_1\cap L_2\in \mathcal{K}\cup {\mathcal{K}}^{\prime }$.

3. $\mid \mathcal{R}\mid =m{m}^{\prime }.$

4. $\mid \mathcal{X}\mid =m{m}^{\prime }\left(q-1\right)+m+m^{\prime }.$

5. If$H\in \mathrm{ℌ}$is a hyperplane and$m_H≔\mid H\cap \mathcal{K}\mid ,m_H^{\prime }≔\mid H\cap {\mathcal{K}}^{\prime }\mid ,$then$\mid H\cap \mathcal{X}\mid =m_H\cdot m_H^{\prime }\left(q-1\right)+m_H+m_H^{\prime }+\left(m-m_H\right)\cdot \left(m^{\prime }-m_H^{\prime }\right).$

6. The linear code$C_{\mathcal{X}}$has length$\mid \mathcal{X}\mid =m{m}^{\prime }\left(q-1\right)+m+m^{\prime }$and dimension k.

If $\mathcal{K}=\mathcal{S}$ and ${\mathcal{K}}^{\prime }={\mathcal{S}}^{\prime }$ then $\mathcal{X}=\mathcal{P}$, hence $C_{\mathcal{X}}=C_{\mathcal{P}}$ is the simplex code of dimension k. As each hyperplane H is contained in $\mathcal{X}$, in such a case every code word has the same weight. Therefore the minimum distance (or, weight) is $d=\mid \mathcal{X}\mid -\mid H\mid =q^{k-1}$.

From [24], Lemma 4 follows

Result If $H_{\mathcal{S}}\subset \mathcal{S}$ and $H_{{\mathcal{S}}^{\prime }}\subset {\mathcal{S}}^{\prime }$ are subspaces of $dim{H}_{\mathcal{S}}=dim\mathcal{S}-1$ and $dim{H}_{{\mathcal{S}}^{\prime }}=dim{\mathcal{S}}^{\prime }-1$, then there exist exactly $q+1$ hyperplanes H with $H_{\mathcal{S}},H_{{\mathcal{S}}^{\prime }}\subset H$ one of which contains $\mathcal{S}$ and one contains ${\mathcal{S}}^{\prime }$.

Let $M_{\mathcal{S}}$ and $M_{{\mathcal{S}}^{\prime }}$ be the type of $\mathcal{K}$ and ${\mathcal{K}}^{\prime }$ with respect to hyperplanes, respectively. Denote $M=M_{\mathcal{S}}\cup \left\{m\right\},M^{\prime }=M_{{\mathcal{S}}^{\prime }}\cup \left\{m^{\prime }\right\},m_o=minM,m_o^{\prime }=min{M}^{\prime },m_1=max{M}_{\mathcal{S}}$ and $m_1^{\prime }=max{M}_{{\mathcal{S}}^{\prime }}$.

Consider the following mapping

$$\iota :M\times M\to \mathbf{N},\iota \left(a,a^{\prime }\right)=a^{\prime }\left(\mathit{aq}+1-m\right)+a\left(1-m^{\prime }\right)+m\cdot m^{\prime }.$$E14

Then the type of $\mathcal{X}$ is $M_{\mathcal{X}}=\left\{\iota \left(a,a^{\prime }\right)|\left(a,a^{\prime }\right)\in M\times M\backslash \left\{\left(m,m^{\prime }\right)\right\}\right\}$ and $max{M}_{\mathcal{X}}=max\left\{\iota \left(m_o,m_o^{\prime }\right),\iota \left(m,m_1^{\prime }\right),\iota \left(m_1,m^{\prime }\right)\right\}$ (cf. [24], Proposition 5 and Lemma 6).

Hence we can determine the weight distribution of $C_{\mathcal{X}}$ once known the types $M_{\mathcal{S}}$ and $M_{{\mathcal{S}}^{\prime }}$ of $\mathcal{K}$ and ${\mathcal{K}}^{\prime }$, respectively.

Proposition 17The code$C_{\mathcal{X}}$is a linear code of length$n=\iota \left(m,m^{\prime }\right)$, dimension k and minimum weight$d=n-max\left\{\iota \left(m_o,m_o^{\prime }\right),\iota \left(m,m_1^{\prime }\right),\iota \left(m_1,m^{\prime }\right)\right\}$.

See [24] Theorem 7.

Since $\mathcal{X}$ generates the projective space $\left(\mathcal{P},\mathfrak{L}\right)$ there exists a basis $\mathcal{B}\subset \mathcal{X}$ of $\left(\mathcal{P},\mathfrak{L}\right)$. Let $\mathcal{X}=\left\{p_1,\dots ,p_n\right\}$ such that $\mathcal{B}=\left\{p_1,\dots ,p_k\right\}$ is a basis. Let $\mathbf{v}\left(\mathcal{X}\right)$ be a system of vectors representing $\mathcal{X}$. For $p_j\in \mathcal{X}$ let $\mathbf{v}\left(p_j\right)={\sum }_{i=1}^k{\gamma }_{\mathit{ij}}\mathbf{v}\left(p_i\right)$ be the vector representing the point $p_j$ with respect to the basis $\mathbf{v}\left(\mathcal{B}\right)$ of $F^k$. Then $G=\left({\gamma }_{\mathit{ij}}\right)$ is a standard generatrix matrix of the code $C_{\mathcal{X}}$.

If ${\mathcal{B}}_{\mathcal{K}}\subset \mathcal{K}$ is a basis of $\mathcal{S}$ and ${\mathcal{B}}_{{\mathcal{K}}^{\prime }}\subset {\mathcal{K}}^{\prime }$ is a basis of ${\mathcal{S}}^{\prime }$ then $\mathcal{B}={\mathcal{B}}_{\mathcal{K}}\cup {\mathcal{B}}_{{\mathcal{K}}^{\prime }}\subset \mathcal{X}$ is a basis of $\left(\mathcal{P},\mathfrak{L}\right)$. With such a basis it is easy to write down the standard generatrix matrix, in particular in the binary case.

For $q=2$, the standard generatrix matrix $\mathcal{G}$ for $C_{\mathcal{X}}$ is shown in [24], p.751.

For the following Examples see Examples 1, 2 and 3 of [24], pp. 751–754.

Example 3 In $\mathit{PG}\left(k-1,q\right),k\ge 5,$ choose and fix an ellipsoid $\mathcal{E}$ in a 3-dimensional subspace $\mathcal{S},$ let ${\mathcal{S}}^{\prime }$ be a complementary subspace of $\mathcal{S}$, set $r≔dim{\mathcal{S}}^{\prime }=k-5$.

The code $C_{\mathcal{X}}$ associated to the ruled set defined by $\mathcal{K}=\mathcal{E},{\mathcal{K}}^{\prime }={\mathcal{S}}^{\prime }$ has length $n=\left(q^2+1\right)\left({\sum }_{i=0}^r{q}^i\right)\left(q-1\right)+q^2+1+{\sum }_{i=0}^r{q}^i=q^{r+3}+{\sum }_{i=0}^{r+1}{q}^i$ and dimension $k=r+5$.

The type of $\mathcal{K}=\mathcal{E}$ is $M_{\mathcal{S}}=\left\{m_o=1,m_1=q+1\right\}$; it holds $t_{m_o}=q^2+1$ and $t_{m_1}=q^3+q$.

The type of ${\mathcal{K}}^{\text{'}}={\mathcal{S}}^{\prime }$ is $M_{{\mathcal{S}}^{\prime }}=\left\{m_0^{\prime }={\sum }_{i=0}^{r-1}{q}^i\right\}$ and $t_{m_o^{\prime }}={\sum }_{i=0}^r{q}^i$.

Then the weight distribution is obtained. There are:

• $q^2+1$ code words of weight $q^{r+3}$,

• ${\sum }_{i=4}^{r+4}{q}^i$ code words of weight $q^{r+3}-q^{r+2}+q^{r+1}$,

• $q^3+q$ code words of weight $q^{r+3}-q^{r+2}$.

This shows that the minimum weight of $C_{\mathcal{X}}$ is $d=q^{r+3}-q^{r+2}$.

For $q=2$ the code $C_{\mathcal{X}}$ is a linear $\left[2^{r+3}+2^{r+2}-1,r+5,2^{r+2}\right]$-code with error-correcting capability $t=2^{r+1}-1$.

For $r=1$ the code $C_{\mathcal{X}}$ is a [23, 6, 8]-code.

Example 4 In $\mathit{PG}\left(5,q\right),q=2^h,$ choose and fix two ovals with their nucleus, $\mathcal{K}\subset \mathcal{S}$ and ${\mathcal{K}}^{\prime }\subset {\mathcal{S}}^{\prime }$, respectively, where $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ are two skew planes.

The code $C_{\mathcal{X}}$ associated to the ruled set defined by $\mathcal{K}$ and ${\mathcal{K}}^{\prime }$ has length $n=\left(q+2\right)\left(q+2\right)\left(q-1\right)+2\left(q+2\right)=q^3+3q^2+2q$ and dimension $k=6$.

There are $\left(\begin{array}{c}q+2\\ 2\end{array}\right)=\frac{1}{2}\left(q^2+3q+2\right)$ lines in $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ meeting $\mathcal{K}$ and ${\mathcal{K}}^{\prime }$ in 2 points and $\frac{1}{2}\left(q^2-q\right)$ lines in $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$ missing $\mathcal{K}$ and ${\mathcal{K}}^{\prime }$, respectively.

We obtain the following weight distribution. There are

• $q^2-q$ code words of weight $q^3+3q^2+q-2$,

• $\frac{1}{2}\left(q^5+q^4-3q^3-q^2+2q\right)$ code words of weight $q^3+2q^2-2$,

• $\frac{1}{4}\left(q^5+5q^4+7q^3-q^2-8q-4\right)$ code words of weight $q^3+2q^2-2q$,

• $\frac{1}{4}\left(q^5-3q^4+3q^3-q^2\right)$ code words of weight $q^3+2q^2-2q-4$,

• $q^2+3q+2$ code words of weight $q^3+q^2-q$.

This shows that the minimum weight of $C_{\mathcal{X}}$ is $d=8$ for $q=2$ and $d=q^3+q^2-q$ for $q\ge 4$.

For $q=2$ the code $C_{\mathcal{X}}$ is a linear [24, 6, 8] -code with error-correcting capability $t=3$.

Example 5 In $\mathit{PG}\left(7,q\right)$ choose and fix two ellipsoids $\mathcal{E}$ and ${\mathcal{E}}^{\prime }$ in two non-intersecting 3-dimensional subspaces $\mathcal{S}$ and ${\mathcal{S}}^{\prime }$, respectively. Then the code $C_{\mathcal{X}}$ related to the ruled set defined by $\mathcal{K}=\mathcal{E},{\mathcal{K}}^{\prime }={\mathcal{E}}^{\prime }$ has length $n=\left(q^2+1\right)\left(q^2+1\right)\left(q-1\right)+2q^2+2=q^5-q^4+2q^3+q+1$ and dimension $k=8$.

There are $q^2+1$ planes $E\subset \mathcal{S}$ with $\mid E\cap \mathcal{E}\mid =1$ and $q^3+q$ planes $E\subset \mathcal{S}$ with $\mid E\cap \mathcal{E}\mid =q+1$.

We obtain the following weight distribution.

There are

• $2\left(q^2+1\right)$ code words of weight $q^5-q^4+q^3$,

• $2\left(q^6-q^5+2q^4-2q^3+q^2-q\right)$ code words of weight $q^5-2q^4+3q^3-q^2$,

• $q^7-q^6+2q^5-2q^4+q^3-q^2$ code words of weight $q^5-2q^4+3q^3-2q^2$,

• $q^5-q^4+2q^3-2q^2+q-1$ code words of weight $q^5-2q^4+2q^3$,

• $2q\left(q^2+1\right)$ code words of weight $q^5-2q^4+2q^3-q^2$.

This shows that the minimum weight of $C_{\mathcal{X}}$ is $d=q^5-2q^4+2q^3-q^2$.

For $q=2$ the code $C_{\mathcal{X}}$ is a linear [35, 8, 12]-code with error-correcting capability $t=5$.

In [24], pp. 752–753 the standard generatrix matrices of the three examples are shown.

7. Conclusions

The close connection between the geometry of the projective varieties, or in general, of suitable subsets of a finite geometry and linear codes through projective systems, certainly still has prospects for interesting developments. This is, on the one hand, because of the elaboration and study of eventually new varieties, and, on the other, for the possibility of constructing linear codes with interesting parameters for the various applications in the communication systems.

Acknowledgments

Dedicated to Professor Hans-Joachim Kroll for his 80th birthday.

Classification:

Mathematics Subject Classification (2020): 94B05, 94B27, 51E20, 51A22