Open access peer-reviewed chapter

# A Geometrical Realisation of Quasi-Cyclic Codes

Written By

Cristina Martinez Ramirez and Alberto Besana

Reviewed: 28 June 2019 Published: 19 November 2019

DOI: 10.5772/intechopen.88288

From the Edited Volume

## Probability, Combinatorics and Control

Edited by Andrey Kostogryzov and Victor Korolev

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## Abstract

We study and enumerate cyclic codes which include generalised Reed-Solomon codes as function field codes. This geometrical approach allows to construct longer codes and to get more information on the parameters defining the codes. We provide a closed formula in terms of Stirling numbers for the number of irreducible polynomials and we relate it with other formulas existing in the literature. Further, we study quasi-cyclic codes as orbit codes in the Grassmannian parameterizing constant dimension codes. In addition, we review Horn’s algorithm and apply it to construct classical codes by their defining ideals.

• cyclic code
• partition
• Grassmannian

## 1. Introduction

Function fields are used ubiquitously in algebraic coding theory for their flexibility in constructions and have produced excellent linear codes. Suitable families of function fields, for example good towers of function fields, have been used to construct families of codes with parameters bound better than the asymptotic bound.

Let q a power of a prime number p. It is well known, that there exists exactly one finite field with q elements which is isomorphic to the splitting field of the polynomial xqx over the prime field Fp. Any other field F of characteristic p contains a copy of Fp. We denote respectively by AnFq and PnFq the affine space and the projective space over Fq. Let Fqx1x2xn be the algebra of polynomials in n variables over Fq.

The encoding of an information word into a k-dimensional subspace is usually known as coding for errors and erasures in random network coding [1]. Namely, let V be an Ndimensional vector space over Fq, a code for an operator channel with ambient space V is simply a non-empty collection of subspaces of V. The collection of subspaces is a code for error correcting errors that happen to send data through an operator channel. The matrix coding the information is parameterised by random variables a1,a2,,an which constitute the letters of an alphabet. Here the operator channel is an abstraction of the operator encountered in random linear network coding, when neither transmitter nor receiver has knowledge of the channel transfer characteristics. The input and output alphabet for an operator channel is the projective geometry. A good code is capable of correcting error and erasures at the output of the operator channel. Thus in order to construct good codes one need to choose a metric consistent with channel errors and search of a set of vectors with given metric properties as a correcting code. The codes considered here are codes for channels whose errors are consistent with the weighted Hamming metric (WHM).

Let C be a non-singular, projective, irreducible curve defined over Fq, as the vanishing locus of a polynomial FFqx0x1x2. We define the number Nq of Fqrational points on the curve to be

Nq=x0x1x2P2FqFx0x1x2=0.

It is a polynomial in q with integer coefficients, whenever q is a prime power.

The number of points C¯Fqr on C over the extensions Fqr of Fq is encoded in an exponential generating series, called the zeta function of C¯:

ZCt=expr=1#C¯Fqrtrr.

Garcia and Stichtenoth analysed the asymptotic behaviour of the number of rational places and the genus in towers of function fields, [2]. From Garcia-Stichtenoth’s second tower one obtains codes over any field Fq where q is an even power of a prime [3].

One of the main problems in coding theory is to obtain non-trivial lower bounds of the number NFi of rational places of towers of function fields Fi/Fqi=1 such that FiFi+1. Suitable families of function fields, for example good towers of function fields, have been used to construct families of codes that beat the Gilbert-Varshamov bound. This paper aims to explore this link for the study and construction of quasi-cyclic codes. For example good codes are obtained for curves of genus 0, they are in fact extended generalised Reed-Solomon codes.

Notation. Let Fq denote the Galois field of q elements and let Fqn denote the vector spaces of all ordered n-tuples over Fq. The Hamming weight of a vector x, denoted by wtx is then number of non-zero entries in x. A linear code C of length n and dimension k over Fq is a k-dimensional subspace of Fqn. Such a code is called nkdq code if its minimum Hamming distance is d. For d a positive integer, α=α1αm is a partition of d into m parts if the αi are positive and decreasing.

## 2. Algebraic geometric codes

Let Fq be a finite field of q elements, where q is a power of a prime. We consider as an alphabet a set P=P1PN of NFq rational points lying on a smooth projective curve C of genus g and degree d defined over the field Fq. If D is a divisor on the curve C, LD is the linear series attached to this divisor with coefficients in the field.

Definition 2.1. Algebraic Geometric Codes (AGC) are constructed by evaluation of the global sections of a line bundle or a vector bundle on the curveCoverNN>gdistinct rational placesP1,,PN. Namely, letFFqbe the function field of the curve,Dthe divisorP1++PNandGa divisor ofFFqof degreesNsuch thatSuppGSuppD=ø. Then the geometric Goppa code associated with the divisorsDandGis defined by

CDG=xP1xPnxLGFqn.

Recall that FqnFq is a cyclic Galois extension and it is finitely generated by unique element αFqn\Fq. α is a primitive element and 1αα2αn1 is a basis of the field extension FqFqα, that is, FqnFqn.

In the sequel, an nkq-code C is a k-dimensional subspace of Fqn.

### 2.1 Generalised Reed-Solomon codes as cyclic codes

Another important family of Goppa codes is obtained considering the normal rational curve (NRC) Cn defined over Fq:

CnFq1ααn:αFq.

Assuming that np=1 are coprime, the set 1αα2αn1 forms a basis of Fqn over Fq, where p is the characteristic of the field. Thus points in the NRC are in correspondence with Fqlinear combinations of the base vectors up to collineation. The Goppa codes of dimension n defined over Cn are constructed by evaluating non-zero polynomials of degree less than n over a sequence α1,,αn of n distinct elements in Fq, if kn, then the map

ϵ:FqxFqn,ffα1αnE1

is injective, since the existence of a non-zero polynomial of degree less than k vanishing on all αi implies n<k by the fundamental theorem of algebra (a non-zero polynomial of degree r with coefficients in a field can have at most r roots). These are just Reed-Solomon codes of parameters nkd over a finite field Fq, with parity check polynomial hx=i=1qxαi, where α is a primitive root of Fq such that αk+1=α+1. Any codeword c0c1cn1 can be expanded into a q-ary k vector with respect to the basis 1ααk1. Construction of generalised Reed-Solomon codes over Fq only employ elements of Fq, hence their lengths are at most q+1. In order to get longer codes, one can make use of elements of an extension of Fq, for instance considering subfield subcodes of Reed-Solomon codes. In this way, one gets cyclic codes. Recall that a linear cyclic code is an ideal in the ring Fqx/xn1 generated by a polynomial gx with roots in the splitting field Fql of xn1, where nql1, ([4]). We shall identify the code with the set of its codewords. A natural question then to ask is how many irreducible polynomials of degree at least 2 are there over the algebraic closure of Fqx. Next theorem expresses this number in terms of Stirling numbers.

Theorem 2.2. Assume thatqn=1, then the number of polynomials of degreen2decomposable into distinct linear factors over a finite fieldFqof arbitrary characteristic a prime numberp, is equal tok=1nqk, whereqkis the falling factorial polynomialqq1qk=k=0nsnkqk, wheresnkis the Stirling number of the first kind (the number of ways to partition a set ofnobjects intoknon-empty subsets), divided by the order of the affine transformation group of the affine lineA1=P1\, that isq2q.

Proof. We need to count all the polynomials fnx in one variable of degree n fixed. We assume that our polynomial fnx decomposes into linear factors, otherwise we work over F¯qx, where F¯q denotes the algebraic closure of the finite field Fq. Since the number of ordered sequences on q symbols is q! and each root is counted with its multiplicity, it follows that the number of monic polynomials with n1 different roots is qq1q2qn+1q2n. Now we observe that polynomials are invariant by the action of automorphisms of the affine line, so we must divide this number by the order of this group which is q2q.

Theorem 2.3. Given a set of integers01n1modulen, there is a setJofkintegers which is a set of roots, that is, there is a polynomialhx=jJxαj, whereαis a generator ofFpmfor some prime numberpandmis the least integer such thatnpm1. The idealhxgenerates inFpmx/xn1is a cyclic linear code of parametersnknk+1.

Proof. Let m be the least integer such that n divides pm1, then g.c.dmp=1. We define an equivalence relation on the set of integers 01n1, by declaring two integers i and j in the range 0in1 to be conjugate module n if psijmodn. This equivalence relation partition the set into cyclotomic cosets. The cyclotomic coset containing j, which we will denote by Ωj, can be described explicitly as the set jpjpk1j, where k is the least positive integer such that pkjjmodn and j is not necessarily the smallest integer in such coset. Denote by In the set consisting of the smallest integers in each cyclotomic coset, then In is a set root, that is, it is a set of k integers in arithmetic progression modulo n whose increment is relatively prime to n. Let d=nk+1, then the polynomial iInxαi defines a cyclic code of parameters nkd.

As an application of Theorem 2.2, given an integer n, we can count the number of cyclic codes of parameters nk for each 0kn and set of roots α1,,αk in the splitting field of xn1, the corresponding polynomial gx=i=1kxαi generates a linear cyclic code in the ring Fqx/xn1. Thus for each 0kn there are exactly qk/q2q cyclic codes.

In the theory of error-correcting codes to a given code CFqn, one assigns another important parameter, the minimum distance d which measures how good the decoding is.

Definition 2.4. The distance between vectorsa=a1a2anandb=b1b2bnin the Weighted Hamming metric (WHM) is defined by a function:

dWHab=i=1nwidaibi¯,

where wi>0, daibi=1 if aibi and daibi=0 if ai=bi. The weight of a vector a in the WHM is wtWHa=dWHa0=i:ai0wi. The value wi and vector w=w1w2wn are called a weight of position i and a vector of weights of positions respectively.

Geometrically a binary vector a1an of length n gives the coordinates of a vertex of a unit cube in n dimensions.

Example 1. Consider the Goppa code defined by the rational functiongx=3x25x+5x32x2+xwhich admits as decomposition into partial fractions the expressionGx5x2x1+3x12. The presence of a double factorx12corresponds to the existence of an eigenspaceEin the vector spaceFqnof multiplicity 2 and thus anαsplitting subspace where the operatorαis just the linear operatorAλI, withλthe eigenvalue associated toEandAis the generator matrix of the code. We recall that anrdimensionalWsubspace isαsplitting ifαiW=Wis invariant under the action of any elementαiin the Galois group of the extensionFqFqα.

### 2.2 Algebraic function field codes

A much greater variety of linear codes is obtained if one uses places of arbitrary degree rather than just places of degree 1. These codes are more naturally described through function field codes. A general viewpoint is that function field codes are certain finite dimensional linear subspaces of an algebraic function field over a finite field as in Goppa’s construction.

In the paper [5], the authors introduce another construction where places of arbitrary degree are allowed. The method consists of choosing two divisors G1 and G2 of an algebraic curve over Fq with G1G2. Then LG1 is a subspace of the vector space LG2 over Fq. If we choose a basis of LG2, then the coordinate vectors of the elements of LG1 form a linear code over Fq of length n=dimLG2 and dimension k=dimLG1. These are known as function field codes and they provide a general perspective on the construction of algebraic-geometry codes [6].

Example 2. We consider as in [7] the Suzuki curveχdefined overFqby the following equationyqy=xq0xqxwithq=2q028andq0=2r. This curve has exactlyq2+1rational places with a single place at infinityPand it is of genusgS=q0q1. We construct a code out of the divisor F=mPandQwhereQ=P1++Pq2is the sum of theq2rational points and the parametermsatisfies the boundm>2g2andgis the genus of the curve.

Observe that the geometric Goppa code CFQ is an Fq-subspace of Fqq2 and its dimension k as an Fqvector space is the dimension of the code. Geometrically, it corresponds to a point in the Grassmannian Gq2,kFq. The set of codewords recognised by the code CFQ admits the following description in terms of monomial ideals in the variables x,y,z,w:

xaybzcwdabcd0aq+bq+q0+cq+2q0+dq+2q0+1d,

where z=x2q0+1 and w=xy2q0z2q0 are elements in the function field FχFqxy over Fq. Moreover, it is a generating set for the linear series LdP associated to the divisor dP.

Theorem 2.5. Cyclic codes are function field codes constructed over the curveCn,mwith affine equationym+xn=1defined over a finite fieldFqofqelements, whereqis a power of a primepandn,mare integer numbers greater or equal than 2.

Proof. Let us assume n is an integer even number, thus n=2ks, with s an integer odd number. We recall that a linear cyclic code is an ideal in the ring Fqx/xn1 generated by a polynomial gx with roots in the splitting field Fql of xn1, where nql1. If we consider the factorisation of the polynomial xn1 over Fpx, we get xn/21xn/2+1=xn/41xn/4+1xn/2+1=xn/2k1xn/2k+1xn/2k11xn/2k1+1xn/2+1. We see that the point P0=α0PFq2 with αn/2=p1 is an Fq2rational place of the affine curve ym=xn/2+1. The other rational places are Pk=β0 with βn/2k=p1,..., P2=β20, P1=10, P0=10 and the place P=0α at . The cyclic code is realised as the algebraic geometric code associated to the divisors D=P0+P1++Pk, G=μP and the parameter μ satisfies the bound μ>2g2, where g is the genus of the curve Cn,m. Note that m is the least integer such that npm1. In particular α is a generator of Fpm.

If n is an integer odd number, by Theorem 2.3, we know there is a set of roots αjjJ, such that α is a generator of Fpm. Now we consider the points Pj=αj0 with jJ and the point P at P=0α=, and the cyclic code is realised as the function field code associated to the divisors D=jJPj and μP.

Remark 2.6. The proof given in theorem gives a realisation of cyclic codes as AG codes constructed over the curve with affine modelym+xn=1. In particular whenm=n=q+1, we cover the codes defined over the Hermitian curve.

Another important family of cyclic codes is obtained considering the roots of the polynomial xn1 over its splitting field. These codes are of great importance in ADN-computing and as they are linear codes, they can be described as function fields. Let α be a primitive element of the underlying vector space over Fq. Since the base field is of characteristic p, xn1 has n different zeroes. Let F¯qx be the extension field containing the nth roots of unity 1,α,,αn1, where αn1+αn2++α+1=0. Moreover the set 1ααn1 constitutes a basis over the prime field Fp, and the field extensions FpnFpx/xn1++x+1 are isomorphic.

Example 3. The polynomialx2+x+1overFpxis irreducible, thus the fieldsFp2Fpx/x2+x+1are isomorphic, and the rootsw,w+1correspond to one place of degree 2 in the extension fieldFpw.

Example 4. We define the polynomialsfx=xn+a1xn1++an, withaiQ, andfxt=fxt. Then, iffis a separable polynomial, then the Galois group offxtoverQtis a regular extension with Galois groupSn.

Observe that Qx1xn is the function field of an n1dimensional projective space Pn1Q over Q. Suppose that z1,,zn are the roots of f in a splitting field of f over Q. Each coefficient ai of xi in f is symmetric in z1,,zn, thus by the theorem on symmetric functions, we can write ai as a symmetric polynomial in z1,,zn with rational coefficients. On the other side, for a permutation σSn, set Eσ=x1zσ1++xnzσn in Qx1xn and fx=σxEσ, where σ runs through all permutations in Sn.

Theorem 2.7. (Hilbert) LetG=Snbe acting onQx1xn. The fieldEofSninvariants isQt1tn, wheretiis theithsymmetric polynomial inx1,,xnandQx1xnhas Galois groupSnoverE. It is the splitting field of the polynomialfx=xnt1xn1++1ntn.

Let F be a finite field such that charFn=1. A non-zero polynomial in F¯xy defines a curve on the plane F¯2. The elliptic curves are curves of the form y2=fx, where fx is a polynomial of degree 3 with coefficients in F¯.

Proposition 2.8. Letn=rsbe a factorisation of an integer positive numberninto irreducible coprime factors and assumer<s, then there is a sequence of field extensionsFqrFqsFqn.

Proof. Consider the map Tn:FnFn

tj=1jσjx1xn,

where σj is the jth elementary symmetric function in the variables xi. Thus tjj=1n, are the coefficients of the equation:

fzt1tn=zn+1t1zn1++1ntn=zx1zx2zxn.

If we apply Theorem 2.7 to the ith elementary symmetric polynomial in the symbols α,αq,αq2,,αqn, we get that the field of Sn invariants of the polynomial fzt1tn contains an extension Fqn of Fq. Moreover, for any divisor r of n, one can consider the field of Sr invariants, and apply Theorem 2.7 to the symbols α,αq2s,,αqrs, where n=rs. Then we get an extension Fqs of Fqr and all its Fq-subspaces are stable under GalFqs/Fqr.

Example 5. Assumen=q+1and we study again the roots of the polynomialxq1in its splitting field. Letξbe a non-trivial n-root of unity, for any divisorrofn, one can consider the symbolsξqr,,ξq,ξ. The field ofSrinvariants of the polynomialfzt1tris the set of solutions to the equation:

xqr++xq+x=ainFqn.E2

In F2n, for any divisor d of n, there are exactly 2d1 solutions to Eq. (2) if n/d is odd and 2d solutions if n/d is even.

Instead of considering r,s divisors of n, we can consider a partition of n into two parts. For example for an integer 0kn, we consider the partition knk of n. Fix two elements g1,g2GLnq of rank k and rank nk. These points correspond to linear transformations Tgi:FqnFqn, i1,2. It is well known that the corresponding points Fqk,qnFqn and Fqnk,qnFqn in the Grassmannians Gk,nFq of kdimensional subspaces and the Grassmannian GnkFq of nk dimensional subspaces respectively are dual subspaces in the underlying vector space Fqn for the Euclidean inner product. Note that the Hamming weight is preserved under invertible linear transformation. This case is of great interest for applications in coding theory, since the corresponding codes with generator matrices G1 and G2 respectively are dual codes. Namely, given a linear nk-code, a parity check matrix for C is an nk×n matrix H of rank nk such that C=xFqn:HcT=0. Then the dual code C is the linear nnk code generated by the parity check matrix of C. There is a right action of the general linear group GLnFq on Gk,nFq:

Gk,nFq×GLnFqGk,nFqE3
UAUA.E4

One can study the orbits of Gk,nFq by the action of any subgroup in the general linear group GLnFq. For example we can study the orbit of any triangle group: the Klein group 2×2, the dihedral group, the alternated groups A4 and A5 or the symmetric group Sn. Take as T the standard shift operator on Fqn, a linear code C is said to be quasi-cyclic of index l or lquasi-cyclic if and only if is invariant under Tl. If l=1, it is just a cyclic code. The quantity mn/l is called the co-index of C. Namely, if we view a codeword c0c1cn1 of C as a polynomial c0+c1x++cn1xn1Fqx, then Tcx=xcxmodxn1.

Example 6. We study the action of a rotation element on the GrassmannianG2,4Fqof lines in a 3-dimensional projective spacePG3q. We apply to any linega rotationτof angleα=2πn, represented by the array of vectors<1,0,0,0cosαsinα,0cosαsinα>. It is easy to see that the orbit code by the composed actionτmwithmdivisor ofnis a quasi-cyclic code of indexmn.

In general, we study generalised Grassmannians or more commonly known as flag varieties. Fix a partition λ=λ1λr of n and let Fλ=FλFq be the variety of partial flags of Fqvector spaces

0=ErEr1E1E0=Fqn

such that dimEi1/Ei=λi. The group GLnFq acts on Fλ in the natural way. Fix an element X0Fλ and denote by Pλ the stabilizer of X0 in G and by Uλ the subgroup of elements gPλ which induces the identity on Ei/Ei+1 for all i=0,1,,n1. Put Lλ=GLλrFq×,×GLλ1Fq, then we have Pλ=Lλ×Uλ.

Proposition 2.9. Let us consider the factorisation ofninto irreducible pairwise coprime factorsn=p1e1p2e2prerwithe1<e2<<er, andλ=e1erbe the partition of exponents. Then there is a flag varietyFλFqof partial flags ofFqvector spaces:

0=ErEr1E1E0=Fqn,

such that dimEi1/Ei=ei.

Proof Observe that the result follows trivially for the case in which n is a prime number e1==er=1. If n=rs factorizes into two irreducible prime factors, the result follow as we have seen above, there is a flag 0=ErEsE0=Fqn and then by induction in r the result follows.

Given a cyclic code over F¯ of length n, its defining set is given by the exponents occurring in gx, where gx is the generator polynomial of the ideal of the code in Fx/xn1.

Let αF¯ be an nth primitive root of unity. The nth cyclotomic polynomial Φnx=1<j<n,jn=1xαjF¯x is the minimal polynomial of α over F. It is monic of degree of the Euler’s totient function φn. It has integer coefficients and it is irreducible over Q. In Qx, we have the factorization into irreducible polynomials:

xn1=dnΦdx.

By Möebius inversion:

Φnx=dnxd1μn/d

In the case of binary codes where q=2, Bezzateev and Shekhunova [8] have obtained that the number of irreducible normalized polynomials I2ml of degree l over F2m satisfy the following equation:

I2ml=1ld/lμd2mld.E5

where μd is the Möebius function:

Let gx equals the least common multiple lcmΦix:αiS, then S is a defining set for C. We will describe the defining set by the exponents occurring in S with S=i1i2il, where i1<<il. A parity check matrix for the code CS is given by:

MS=αi10αi11αi1n1αi20αi21αi2n1αir0αir1αirn1)

The code CFn is obtained as the subfield subcode of C¯:

C=cFn:MRcT=0.

Given a triple IJK of subsets of 1n of the same cardinality r, we associate to them partitions λ,μ and ν as follows. Let I=i1<<ir1n, then the corresponding partition is defined as λ=irrr11, and respectively for J,K. We call the triple IJK admissible for the Horn problem, if the corresponding triple of partitions λμγ occurs as eigenvalues of a triple of Hermitian r by r matrices, with the third one the sum of the first two.

We describe Horn’s inductive procedure to produce set of triples IJK01n.

Urn=IJKiIi+jJj=kKk+rr+1/2,
Trn={IJKUrnforallp<randallFGHTpr,
fFif+gGjghHkh+pp+1/2}.

Example 7. Let us consider the triple of subsets

IJK=1,3,51,3,52,4,6

and the corresponding triple of partitions λμν=2,1,02,1,03,2,1 arises from the triple of diagonal 3 by 3 matrices with diagonal entries 2,1,0,1,0,2 and 3,1,2.

Lemma 2.10. For any tripleIJKadmissible for the Horn problem, the polynomials defined byfx=iIxαi,gx=jJxαj,andhx=kKxαkgenerate a cyclic code of lengthn=ir+jr+krmodpandk=r, wherer=I+J+Kandpis the characteristic of the fieldF.

Proof. The cyclic code generated by fx coincides with the cyclic code generated by lcmmix:αiiI and respectively for gx and hx the polynomials lcmmjx:αjjJ and lcmmkx:αkkK. It is the cyclic code generated by the minimal polynomial of αir+jr+kr.

Remark 2.11. We see that Horn’s algorithm is relevant since some classical code constructions can be seen as ideals in a finite dimensional commutative semi simple algebra over a finite fieldFqwithq=prelements andpa prime number as in example (3).

Lemma 2.12. The family of cyclic codes obtained by considering the roots of the polynomialxn1over its splitting fields are indeed AG codes arising from genus 0 curves, and by Riemann-Roch theorem, their parameters satisfy the bounddn+1k, wheredis the minimum distance.

Proof Let F¯qx be the extension field containing the nth roots of unity 1,α,,αn1, where αn1+αn2++α+1=0. Moreover the set 1ααn1 constitutes a basis over the prime field Fp, and the field extensions FpnFpx/xn1++x+1 are isomorphic.

### 2.3 Generating functions of conjugacy classes in a group

The automorphism group of the projective line PFq is the projective linear group PGL2q. Any finite subgroup APGL2q defines a kuniform Cayley (sum) hypergraph ΓkA whose vertices are the generating ktuples of A and the edges are kelement sets x1xkGk represented by random variables x1,,xk. In particular, if fz is the ordinary generating function that enumerates A, that is, number of conjugacy classes in A, then 11fz is the ordinary generating function enumerating sequences of k elements in A. If G is an abelian group, then x1++xkA. In general, we will consider k-arcs in ΓA which represent casual connections between the variables. Applications are known in statistics, for example the multinomial experiment consists of n identical independent trials, and there are k possible outcomes (classes, categories or cells) to each trial and the cell counts n1,n2,,nk are the random variables, the number of observations that fall into each of the kcategories.

Definition 2.13. InPGn1qakrarc is a set ofkpoints anyrof which form a basis forFqn, or in other words,r1of them but notrare collinear.

Consider the normal rational curve over Fq:

V1nFq1xx2xnxFq

is a (q + 1)-arc in the n-dimensional projective space PGnq.

We see that if qn, the NRC is a basis of a q-dimensional projective subspace, that is, a PGqnq. So we can enumerate how many NRC’s are there in a PGnq. The answer is ϕqnq, the number of ways of choosing such a set of points in a particular q-space. If qn+2, the NRC is an example of a (q + 1)-arc. It contains q + 1 points, and every set of n+1 points are linearly independent.

### 2.4 Conclusion

The problem of considering finite subgroups and conjugacy classes in PGL2q the automorphism group of the projective line can be generalised to that of finite subgroups in PGLnq, the collineation group of the normal rational curve.

## Acknowledgments

This research has been partially supported by the COST Action IC1104 and the project ARES (Team for Advanced Research on Information Security and Privacy. Funded by Ministry of Economy and Competitivity).

## References

1. 1. Kötter R, Kschischang FR. Coding for errors and erasures in random network coding. IEEE Transactions on Information Theory. 2008;54(8)
2. 2. Garcia A, Stichtenoth H. On the asymptotic behaviour of some towers of function fields over finite fields. Journal of Number Theory. 1996;61(2):248-273
3. 3. Geil O, Martin S, Martínez-Peas U, Ruano D. Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower. In: Proceedings of 21st Conference on Applications of Computer Algebra
4. 4. Bezzateev SV, Shekhunova NA. Subclass of cyclic Goppa codes. IEEE Transactions on Information Theory. Nov. 2013;59(11)
5. 5. Niederreiter H, Xing C, Lam KY. A new construction of algebraic-geometry codes. Applicable Algebra in Engineering, Communication and Computing. 1999;9:373-381
6. 6. Hachenberger D, Niederreiter H, Xing C. Function field codes. Applicable Algebra in Engineering, Communication and Computing. 2008;19:201-211
7. 7. Couvreur A, Márquez I, Pellikaan R. A polynomial time attack against algebraic geometry code based public key cryptosystems. arXiv:1401.6025
8. 8. Bezzateev SV, Shekhunova N. Class of generalized Goppa codes perfect in weighted hamming metric. Des. Codes Criptogr. 2013;66:391-399

Written By

Cristina Martinez Ramirez and Alberto Besana

Reviewed: 28 June 2019 Published: 19 November 2019