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A Geometrical Realisation of Quasi-Cyclic Codes

Written By

Cristina Martinez Ramirez and Alberto Besana

Submitted: March 29th, 2019 Reviewed: June 28th, 2019 Published: November 19th, 2019

DOI: 10.5772/intechopen.88288

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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 qa power of a prime number p. It is well known, that there exists exactly one finite field with qelements which is isomorphic to the splitting field of the polynomial xqxover the prime field Fp. Any other field Fof characteristic pcontains a copy of Fp. We denote respectively by AnFqand PnFqthe affine space and the projective space over Fq. Let Fqx1x2xnbe the algebra of polynomials in nvariables 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 Vbe an Ndimensional vector space over Fq, a code for an operator channel with ambient space Vis 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,,anwhich 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 Cbe a non-singular, projective, irreducible curve defined over Fq, as the vanishing locus of a polynomial FFqx0x1x2. We define the number Nqof Fqrational points on the curve to be


It is a polynomial in qwith integer coefficients, whenever qis a prime power.

The number of points C¯Fqron Cover the extensions Fqrof Fqis encoded in an exponential generating series, called the zeta function of C¯:


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 Fqwhere qis 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 NFiof rational places of towers of function fields Fi/Fqi=1such 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 Fqdenote the Galois field of qelements and let Fqndenote the vector spaces of all ordered n-tuples over Fq. The Hamming weight of a vector x, denoted by wtxis then number of non-zero entries in x. A linear code Cof length nand dimension kover Fqis a k-dimensional subspace of Fqn. Such a code is called nkdqcode if its minimum Hamming distance is d. For da positive integer, α=α1αmis a partition of dinto mparts if the αiare positive and decreasing.


2. Algebraic geometric codes

Let Fqbe a finite field of qelements, where qis a power of a prime. We consider as an alphabet a set P=P1PNof NFqrational points lying on a smooth projective curve Cof genus gand degree ddefined over the field Fq. If Dis a divisor on the curve C, LDis 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


Recall that FqnFqis a cyclic Galois extension and it is finitely generated by unique element αFqn\Fq. αis a primitive element and 1αα2αn1is a basis of the field extension FqFqα, that is, FqnFqn.

In the sequel, an nkq-code Cis 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) Cndefined over Fq:


Assuming that np=1are coprime, the set 1αα2αn1forms a basis of Fqnover Fq, where pis 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 ndefined over Cnare constructed by evaluating non-zero polynomials of degree less than nover a sequence α1,,αnof ndistinct elements in Fq, if kn, then the map


is injective, since the existence of a non-zero polynomial of degree less than kvanishing on all αiimplies n<kby the fundamental theorem of algebra (a non-zero polynomial of degree rwith coefficients in a field can have at most rroots). These are just Reed-Solomon codes of parameters nkdover a finite field Fq, with parity check polynomial hx=i=1qxαi, where αis a primitive root of Fqsuch that αk+1=α+1. Any codeword c0c1cn1can be expanded into a q-ary kvector with respect to the basis 1ααk1. Construction of generalised Reed-Solomon codes over Fqonly 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/xn1generated by a polynomial gxwith roots in the splitting field Fqlof 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 fnxin one variable of degree nfixed. We assume that our polynomial fnxdecomposes into linear factors, otherwise we work over F¯qx, where F¯qdenotes the algebraic closure of the finite field Fq. Since the number of ordered sequences on qsymbols is q!and each root is counted with its multiplicity, it follows that the number of monic polynomials with n1different 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 mbe the least integer such that ndivides pm1, then g.c.dmp=1. We define an equivalence relation on the set of integers 01n1, by declaring two integers iand jin the range 0in1to be conjugate module nif 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 kis the least positive integer such that pkjjmodnand jis not necessarily the smallest integer in such coset. Denote by Inthe set consisting of the smallest integers in each cyclotomic coset, then Inis a set root, that is, it is a set of kintegers in arithmetic progression modulo nwhose increment is relatively prime to n. Let d=nk+1, then the polynomial iInxαidefines 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 nkfor each 0knand set of roots α1,,αkin the splitting field of xn1, the corresponding polynomial gx=i=1kxαigenerates a linear cyclic code in the ring Fqx/xn1. Thus for each 0knthere are exactly qk/q2qcyclic codes.

In the theory of error-correcting codes to a given code CFqn, one assigns another important parameter, the minimum distance dwhich 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:


where wi>0, daibi=1if aibiand daibi=0if ai=bi. The weight of a vector ain the WHM is wtWHa=dWHa0=i:ai0wi. The value wiand vector w=w1w2wnare called a weight of position iand a vector of weights of positions respectively.

Geometrically a binary vector a1anof length ngives the coordinates of a vertex of a unit cube in ndimensions.

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 G1and G2of an algebraic curve over Fqwith G1G2. Then LG1is a subspace of the vector space LG2over Fq. If we choose a basis of LG2, then the coordinate vectors of the elements of LG1form a linear code over Fqof length n=dimLG2and 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 divisorF=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 CFQis an Fq-subspace of Fqq2and its dimension kas 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 CFQadmits the following description in terms of monomial ideals in the variables x,y,z,w:


where z=x2q0+1and w=xy2q0z2q0are elements in the function field FχFqxyover Fq. Moreover, it is a generating set for the linear series LdPassociated 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 nis an integer even number, thus n=2ks, with san integer odd number. We recall that a linear cyclic code is an ideal in the ring Fqx/xn1generated by a polynomial gxwith roots in the splitting field Fqlof xn1, where nql1. If we consider the factorisation of the polynomial xn1over 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=α0PFq2with αn/2=p1is an Fq2rational place of the affine curve ym=xn/2+1. The other rational places are Pk=β0with βn/2k=p1,..., P2=β20, P1=10, P0=10and the place P=0αat . The cyclic code is realised as the algebraic geometric code associated to the divisors D=P0+P1++Pk, G=μPand the parameter μsatisfies the bound μ>2g2, where gis the genus of the curve Cn,m. Note that mis the least integer such that npm1. In particular αis a generator of Fpm.

If nis an integer odd number, by Theorem 2.3, we know there is a set of roots αjjJ, such that αis a generator of Fpm. Now we consider the points Pj=αj0with jJand the point Pat P=0α=, and the cyclic code is realised as the function field code associated to the divisors D=jJPjand 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 xn1over 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 αbe a primitive element of the underlying vector space over Fq. Since the base field is of characteristic p, xn1has ndifferent zeroes. Let F¯qxbe the extension field containing the nthroots of unity 1,α,,αn1,where αn1+αn2++α+1=0. Moreover the set 1ααn1constitutes a basis over the prime field Fp, and the field extensions FpnFpx/xn1++x+1are 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 Qx1xnis the function field of an n1dimensional projective space Pn1Qover Q. Suppose that z1,,znare the roots of fin a splitting field of fover Q. Each coefficient aiof xiin fis symmetric in z1,,zn, thus by the theorem on symmetric functions, we can write aias a symmetric polynomial in z1,,znwith rational coefficients. On the other side, for a permutation σSn, set Eσ=x1zσ1++xnzσnin Qx1xnand fx=σxEσ, where σruns 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 Fbe a finite field such that charFn=1. A non-zero polynomial in F¯xydefines a curve on the plane F¯2. The elliptic curves are curves of the form y2=fx, where fxis 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


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


If we apply Theorem 2.7 to the ithelementary symmetric polynomial in the symbols α,αq,αq2,,αqn, we get that the field of Sninvariants of the polynomial fzt1tncontains an extension Fqnof Fq. Moreover, for any divisor rof n, one can consider the field of Srinvariants, and apply Theorem 2.7 to the symbols α,αq2s,,αqrs, where n=rs. Then we get an extension Fqsof Fqrand 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:


In F2n, for any divisor dof n, there are exactly 2d1solutions to Eq. (2) if n/dis odd and 2dsolutions if n/dis even.

Instead of considering r,sdivisors of n, we can consider a partition of ninto two parts. For example for an integer 0kn, we consider the partition knkof n. Fix two elements g1,g2GLnqof rank kand rank nk. These points correspond to linear transformations Tgi:FqnFqn, i1,2. It is well known that the corresponding points Fqk,qnFqnand Fqnk,qnFqnin the Grassmannians Gk,nFqof kdimensional subspaces and the Grassmannian GnkFqof nkdimensional subspaces respectively are dual subspaces in the underlying vector space Fqnfor 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 G1and G2respectively are dual codes. Namely, given a linear nk-code, a parity check matrix for Cis an nk×nmatrix Hof rank nksuch that C=xFqn:HcT=0. Then the dual code Cis the linear nnkcode generated by the parity check matrix of C. There is a right action of the general linear group GLnFqon Gk,nFq:


One can study the orbits of Gk,nFqby 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 A4and A5or the symmetric group Sn. Take as Tthe standard shift operator on Fqn, a linear code Cis said to be quasi-cyclic of index lor lquasi-cyclic if and only if is invariant under Tl. If l=1, it is just a cyclic code. The quantity mn/lis called the co-index of C. Namely, if we view a codeword c0c1cn1of Cas 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=λ1λrof nand let Fλ=FλFqbe the variety of partial flags of Fqvector spaces


such that dimEi1/Ei=λi. The group GLnFqacts on Fλin the natural way. Fix an element X0Fλand denote by Pλthe stabilizer of X0in Gand by Uλthe subgroup of elements gPλwhich induces the identity on Ei/Ei+1for 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:


such that dimEi1/Ei=ei.

ProofObserve that the result follows trivially for the case in which nis a prime number e1==er=1. If n=rsfactorizes into two irreducible prime factors, the result follow as we have seen above, there is a flag 0=ErEsE0=Fqnand then by induction in rthe result follows.

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

Let αF¯be an nthprimitive root of unity. The nthcyclotomic polynomial Φnx=1<j<n,jn=1xαjF¯xis the minimal polynomial of αover 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:


By Möebius inversion:


In the case of binary codes where q=2, Bezzateev and Shekhunova [8] have obtained that the number of irreducible normalized polynomials I2mlof degree lover F2msatisfy the following equation:


where μdis the Möebius function:


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


The code CFnis obtained as the subfield subcode of C¯:


Given a triple IJKof subsets of 1nof the same cardinality r, we associate to them partitions λ,μand νas follows. Let I=i1<<ir1n, then the corresponding partition is defined as λ=irrr11, and respectively for J,K. We call the triple IJKadmissible for the Horn problem, if the corresponding triple of partitions λμγoccurs as eigenvalues of a triple of Hermitian rby rmatrices, with the third one the sum of the first two.

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


Example 7.Let us consider the triple of subsets


and the corresponding triple of partitions λμν=2,1,02,1,03,2,1arises from the triple of diagonal 3 by 3 matrices with diagonal entries 2,1,0,1,0,2and 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 fxcoincides with the cyclic code generated by lcmmix:αiiIand respectively for gxand hxthe polynomials lcmmjx:αjjJand 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.

ProofLet F¯qxbe the extension field containing the nthroots of unity 1,α,,αn1,where αn1+αn2++α+1=0. Moreover the set 1ααn1constitutes a basis over the prime field Fp, and the field extensions FpnFpx/xn1++x+1are isomorphic.

2.3 Generating functions of conjugacy classes in a group

The automorphism group of the projective line PFqis the projective linear group PGL2q. Any finite subgroup APGL2qdefines a kuniform Cayley (sum) hypergraph ΓkAwhose vertices are the generating ktuples of Aand the edges are kelement sets x1xkGkrepresented by random variables x1,,xk. In particular, if fzis the ordinary generating function that enumerates A, that is, number of conjugacy classes in A, then 11fzis the ordinary generating function enumerating sequences of kelements in A. If Gis an abelian group, then x1++xkA. In general, we will consider k-arcs in ΓAwhich represent casual connections between the variables. Applications are known in statistics, for example the multinomial experiment consists of nidentical independent trials, and there are kpossible outcomes (classes, categories or cells) to each trial and the cell counts n1,n2,,nkare 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:


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+1points are linearly independent.

2.4 Conclusion

The problem of considering finite subgroups and conjugacy classes in PGL2qthe automorphism group of the projective line can be generalised to that of finite subgroups in PGLnq, the collineation group of the normal rational curve.



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).


  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

Submitted: March 29th, 2019 Reviewed: June 28th, 2019 Published: November 19th, 2019