On Goppa Codes and Weierstrass Gaps at Several Points

We generalize results of Homma and Kim [J. Pure Appl. Algebra Vol. 162, (2001), pp. 273–290] concerning an improvement on the Goppa bound on the minimum distance of certain Goppa codes.This paper was written while the second author was visiting The University of Valladolid (Dpto. Algebra, Geometría y Topología; Fac. de Ciencias) supported by the Grant SB2000-0225 from the Secretaria de Estado de Educacíon y Universidades del Ministerio de Educacíon, Cultura y Deportes de España.     

低密度奇偶校验码的构造

低密度奇偶校验码(LDPC)最早是由Gallager于1962年提出.它们是线性分组码,其比特错误率极大地接近香农界.1995年Mackay和Neal发掘了LDPC码的新应用后,LDPC码引起了人们的广泛关注.本文利用组合结构给出一些新的LDPC码:利用可分组设计构造一类Tanner图中不含四长圈的正则LDPC码.     

Weierstrass Pairs and Minimum Distance of Goppa Codes

We prove that elements of the Weierstrassgap set of a pair of points may be used to define a geometricGoppa code which has minimum distance greater than the usuallower bound. We determine the Weierstrass gap set of a pair ofany two Weierstrass points on a Hermitian curve and use thisto increase the lower bound on the minimum distance of particularcodes defined using a linear combination of the two points.     

Subcodes of the Projective Generalized Reed-Muller Codes Spanned by Minimum-Weight Vectors

We use methods of Mortimer [19] to examine the subcodes spanned by minimum-weight vectors of the projective generalized Reed-Muller codes and their duals. These methods provide a proof, alternative to a dimension argument, that neither the projective generalized Reed-Muller code of order r and of length over the finite field F _q of prime-power order q, nor its dual, is spanned by its minimum-weight vectors for 0<r<m–1 unless q is prime. The methods of proof are the projective analogue of those developed in [17], and show that the codes spanned by the minimum-weight vectors are spanned over F _q by monomial functions in the m variables. We examine the same question for the subfield subcodes and their duals, and make a conjecture for the generators of the dual of the binary subfield subcode when the order r of the code is 1.     

Lower Bounds on the State Complexity of Geometric Goppa Codes

We reinterpret the state space dimension equations for geometric Goppa codes. An easy consequence is that if deg then the state complexity of is equal to the Wolf bound. For deg , we use Clifford's theorem to give a simple lower bound on the state complexity of . We then derive two further lower bounds on the state space dimensions of in terms of the gonality sequence of . (The gonality sequence is known for many of the function fields of interest for defining geometric Goppa codes.) One of the gonality bounds uses previous results on the generalised weight hierarchy of and one follows in a straightforward way from first principles; often they are equal. For Hermitian codes both gonality bounds are equal to the DLP lower bound on state space dimensions. We conclude by using these results to calculate the DLP lower bound on state complexity for Hermitian codes.     

True Dimension of Some Binary Quadratic Trace Goppa Codes

We compute in this paper the true dimension over of Goppa Codes (L, g) defined by the polynomial proving, this way, a conjecture stated in [14,16].     

The New Minimum Distance Bounds of Goppa Codes and Their Decoding

A couple of new lower bounds of the minimum distance of Goppa codes is derived, using an extended field code for a Goppa code which contains the Goppa code as its subfield-subcode. Also presented are procedures for both error-only and error-and-erasure decoding for Goppa codes up to the new lower bounds, based on the Berlekamp-Massey algorithm and the Feng-Tzeng multisequence shift-register synthesis algorithms which have been used for decoding cyclic codes up to the BCH and HT(Hartmann-Tzeng) bounds.     

Proof of Conjectures on the True Dimension of Some Binary Goppa Codes

There is a classical lower bound on the dimension of a binary Goppa code. We survey results on some specific codes whose dimension exceeds this bound, and prove two conjectures on the true dimension of two classes of such codes.Part of this work has been presented at the Sixth International Conference on Finite Fields and Applications, Oaxaca, Mexico, May 2001.AMS classification: 94B65     

A Subclass of Binary Goppa Codes With Improved Estimation of the Code Dimension

A new bound for the dimension of binary Goppa codes belonging to a specific subclass is given. This bound improves the well-known lower bound for Goppa codes.     

Optimal Subcodes of Second Order Reed-Muller Codes andMaximal Linear Spaces of Bivectors of Maximal Rank

There are exactlytwo non-equivalent [32,11,12]-codes in the binaryReed-Muller code which contain and have the weight set {0,12,16,20,32}. Alternatively,the 4-spaces in the projective space over the vector space for which all points have rank 4 fall into exactlytwo orbits under the natural action of PGL(5) on .     

On automorphism groups of certain Goppa codes

Goppa codes are linear codes arising from algebraic curves over finite fields. Sufficient conditions are given ensuring that all automorphisms of a Goppa code are inherited from the automorphism group of the curve. In some cases, these conditions are also necessary. The cases of curves with large automorphism groups, notably the Hermitian and the Deligne-Lusztig curves, are investigated in detail. This research was performed within the activity of GNSAGA of the Italian INDAM, with the financial support of the Italian Ministry MIUR, project “Strutture geometriche, combinatorica e loro applicazioni”, PRIN 2006–2007.     

Non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

Designs, Codes and Cryptography - In this paper, we construct a family of non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes. Moreover, we present a...     

On the duals of geometric Goppa codes from norm-trace curves

In this paper we study the dual codes of a wide family of evaluation codes on norm-trace curves. We explicitly find out their minimum distance and give a lower bound for the number of their minimum-weight codewords. A general geometric approach is performed and applied to study in particular the dual codes of one-point and two-point codes arising from norm-trace curves through Goppaʼs construction, providing in many cases their minimum distance and some bounds on the number of their minimum-weight codewords. The results are obtained by showing that the supports of the minimum-weight codewords of the studied codes obey some precise geometric laws as zero-dimensional subschemes of the projective plane. Finally, the dimension of some classical two-point Goppa codes on norm-trace curves is explicitely computed.     

On algebras admitting a complete set of near weights,evaluation codes,and Goppa codes

In 1998 Høholdt, van Lint and Pellikaan introduced the concept of a “weight function” defined on a \({\mathbb{F}_q}\)-algebra and used it to construct linear codes, obtaining among them the algebraic geometry (AG) codes supported on one point. Later, in 1999, it was proved by Matsumoto that all codes produced using a weight function are actually AG codes supported on one point. Recently, “near weight functions” (a generalization of weight functions), also defined on a \({\mathbb{F}_q}\)-algebra, were introduced to study codes supported on two points. In this paper we show that an algebra admits a set of m near weight functions having a compatibility property, namely, the set is a “complete set”, if and only if it is the ring of regular functions of an affine geometrically irreducible algebraic curve defined over \({\mathbb{F}_q}\) whose points at infinity have a total of m rational branches. Then the codes produced using the near weight functions are exactly the AG codes supported on m points. A bound for the minimum distance of these codes is presented with examples which show that in some situations it compares better than the usual Goppa bound.     

On the Orthogonality of Geometric Codes

We solve some problems concerning the orthogonality of geometric codes associated with sets of i- and j-dimensional subspaces of PG(n, q). Various applications are found, and we discuss all the interesting cases in small dimensional spaces.     

On the Extendability of Linear Codes

R. Hill and P. Lizak (1995, in “Proc. IEEE Int. Symposium on Inform. Theory, Whistler, Canada,” pp. 345) proved that every [n, k, d]_q code with gcd(d, q)=1 and with all weights congruent to 0 or d (modulo q) is extendable to an [n+1, k, d+1]_q code with all weights congruent to 0 or d+1 (modulo q). We give another elementary geometrical proof of this theorem, which also yields the uniqueness of the extension.     

Cyclicity of the unramified Iwasawa module

For a number field k and a prime number p, let k _?? be the cyclotomic Z _p -extension of k with finite layers k _n . We study the finiteness of the Galois group X _?? over k _?? of the maximal abelian unramified p-extension of k _?? when it is assumed to be cyclic. We then focus our attention to the case where p?=?2 and k is a real quadratic field and give the rank of the 2-primary part of the class group of k _n . As a consequence, we determine the complete list of real quadratic number fields for which X _?? is cyclic non trivial. We then apply these results to the study of Greenberg??s conjecture for infinite families of real quadratic fields thus generalizing previous results obtained by Ozaki and Taya.     

On the Weight Hierarchy of Product Codes

Let V and W be codes and let C = V W be the product code of V and W. In [6] Wei and Yang, conjectured a formula for the generalized Hamming weights of V W in terms of those of V and W provided that both V and W satisfy the chain condition. Recently the conjecture has been proved by Schaathun [4]. In this paper we generalize the formula to a product code with more than two components.     

On Perfect Codes and Related Concepts

The concept of diameter perfect codes, which seems to be a natural generalization of perfect codes (codesattaining the sphere–packing bound) is introduced. This was motivated by the code–anticode bound of Delsartein distance regular graphs. This bound in conjunction with the recent complete solutions of diametric problems in the Hamming graph _q(n) and the Johnson graph J(n,k)gives a sharpening of the sphere–packing bound. Some necessaryconditions for the existence of diameter perfect codes are given.In the Hamming graph all diameter perfect codes over alphabetsof prime power size are characterized. The problem of tilingof the vertex set of J(n,k) with caps (and maximalanticodes) is also examined.