New self-dual codes over GF(4) with the highest known minimumweights

The purpose of this correspondence is to construct new Hermitian self-dual codes over GF(4) of lengths 22, 24, 26, 32, and 34 which have the highest known minimum weights. In particular, for length 22, we construct eight new extremal self-dual [22,11,8] codes over GF(4) which do not have a nontrivial automorphism of odd order. The existence of such codes has been left open since 1991 by Huffman     

Circulant based extremal additive self-dual codes over GF(4)

It is well known that the problem of finding stabilizer quantum-error-correcting codes (QECC) is transformed into the problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product. Our purpose is to classify the extremal additive circulant self-dual codes of lengths up to 15, and construct good codes for lengths 16/spl les/n/spl les/27. We also classify the extremal additive 4-circulant self-dual codes of lengths 4,6,8,12,14, and 16 and most codes of length 10, and construct good codes of even lengths up to 22. Furthermore, we classify the extremal additive bordered 4-circulant self-dual codes of lengths 3,5,7,9,11,13,15, and 17, and construct good codes for lengths 19,21,23, and 25. We give the current status of known extremal (or optimal) additive self-dual codes of lengths 12 to 27.     

环F2+uF2+…+uk-1F2上常循环自对偶码

最近,剩余类环上的常循环码及常循环自对偶码引起了编码学者的极大关注.本文首先利用一些相关的线性码,建立了一类特殊有限链环上长为N的常循环自对偶码的一般理论,利用其结果给出了该环上长为N的(1+uλ)-常循环自对偶码存在的充分条件,得到了该环上长为N的一些常循环自对偶码,并给出了其生成多项式.     

环F2+uF2+…+uk-1F2上长为2S的(1+u)-常循环码的距离分布

研究码字的距离分布是编码理论的一个重要研究方向。该文定义了环R=F₂+uF₂+…+u^k-1F₂上的Homogeneous重量,研究了环R上长为2^S的(1+u)-常循环码的Hamming距离和Homogeneous距离。使用了有限环和域的理论,给出了环R上长为2^S的(1+u)-常循环码和循环自对偶码的结构和码字个数。并利用该常循环码的结构,确定了环R上长为2^S的(1+u)-常循环码的Hamming距离和Homogeneous距离分布。     

Shadow bounds for self-dual codes

Conway and Sloane (1990) have previously given an upper bound on the minimum distance of a singly-even self-dual binary code, using the concept of the shadow of a self-dual code. We improve their bound, finding that the minimum distance of a self-dual binary code of length n is at most 4[n/24]+4, except when n mod 24=22, when the bound is 4[n/24]+6. We also show that a code of length a multiple of 24 meeting the bound cannot be singly-even. The same technique gives similar results for additive codes over GF(4) (relevant to quantum coding theory)     

Optimal double circulant self-dual codes over F4

Optimal double circulant self-dual codes over F₄ have been found for each length n⩽40. For lengths n⩽14, 20, 22, 24, 28, and 30, these codes are optimal self-dual codes. For length 26, the code attains the highest known minimum weight. For n⩾32, the codes presented provide the highest known minimum weights. The [36,18,12] self-dual code improves the lower bound on the highest minimum weight for a [36,18] linear code     

Cyclic codes and self-dual codes over F2+uF2

We introduce linear cyclic codes over the ring F₂+uF ₂={0,1,u,u¯=u+1}, where u²=0 and study them by analogy with the Z₄ case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z₄-codes. Unlike the Z₄ case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map     

基于GF(qN)上秩距离码的校验矩阵的验证方案

J.Stern(1996)在“公钥验证的一个新范例”中基于GF（2）上纠错码的校验矩阵提出了一验证方案。该文基于GF（q^N)(q为素数）上秩距离码的校验矩阵提出了一新的验证方案,将J.Stern的方案中对秘密数据s的重量限制改为对s秩的限制;证明了在随机预方模型中给出的协议是零知识交互证明,并显示出通过参数的适当选取,此方案比J.Stern的方案更安全。     

A Varshamov-Gilbert bound for a class of formally self-dual codesand related quantum codes

It is proved that a class of q-ary (2n,n) formally self-dual codes obtained from symmetric matrices over GF (q), contains codes that meet the Varshamov-Gilbert bound. The codes are self-dual with respect to the symplectic inner product and yield quantum codes encoding one state with n q-ary qubits and having minimum distance proportional to n     

环Z4上自对偶码的构造

利用有限环Z₄+vZ₄（其中v²=1）上自对偶码,给出了一种构造Z₄上自对偶码的方法.引入了（Z₄+vZ₄）ⁿ到Z₄²ⁿ的保距Gray映射,给出了Z₄+vZ₄上自对偶码的性质,证明了Z₄+vZ₄上长为n的自对偶码的Gray像是Z₄上长为2n的自对偶码,由此构造了Z₄上一些极优的类型I与类型Ⅱ自对偶码.     

New multilevel codes over GF(q)

Set partitioning is applied to multidimensional signal spaces over GF(q), i.e., GFⁿ¹(q) (n1⩽q ), and it is shown how to construct both multilevel block codes and multilevel trellis codes over GF(q). Multilevel (n, k, d) block codes over GF(q) with block length n, number of information symbols k, and minimum distance d_min⩾d are presented. These codes use Reed-Solomon codes as component codes. Longer multilevel block codes are also constructed using q-ary block codes with block length longer than q+1 as component codes. Some quaternary multilevel block codes are presented with the same length and number of information symbols as, but larger distance than, the best previously known quaternary one-level block codes. It is proved that if all the component block codes are linear. the multilevel block code is also linear. Low-rate q-ary convolutional codes, word-error-correcting convolutional codes, and binary-to-q-ary convolutional codes can also be used to construct multilevel trellis codes over GF(q) or binary-to-q-ary trellis codes     

The children of the (32, 16) doubly even codes

Generator matrices, weight distributions, and automorphism groups are given for all (26,13,6), (28,14,6), and (30,15,6) binary self-dual codes. These results are obtained from the earlier enumeration made by J. H. Conway and the author of all (32,16) self-dual doubly even codes.     

Quantum error correction via codes over GF(4)

The problem of finding quantum error correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits     

Shadow codes and weight enumerators

The technique of using shadow codes to build larger self-dual codes is extended to codes over arbitrary fields. It is shown how to build the codes and how to determine the new weight enumerator as well. For codes over fields equipped with a square root of -1 and not of characteristic 2, a self-dual code of length n+2 can be built from a self-dual code of length n; for codes over a field without a square root of -1 and not of characteristic 2 a self-dual code of length n+4 is built from a self-dual code of length n; and for codes over fields of characteristic 2 the length of the new self-dual code depends on the presence of the all-one vector in the subcode chosen. In certain cases using the subcode of vectors orthogonal to the all-one vector, the new weight enumerator can be calculated directly from the original weight enumerator. Specific examples of the technique are illustrated for codes over F₃, F₄, and F₅     

一种基于BIBD的量子LDPC码构造新方法

利用均衡不完全区组设计(Balance Imcomplete Block Designs,BIBD)的半结构化低密度奇偶校验(Low Density Parity Check,LDPC)码设计方法,该文提出了一种获得自对偶CSS(Calderbank-Shor-Steane)式的量子LDPC码的校验矩阵的新构造方法。由于构造出的量子码校验矩阵稀疏,有且仅有一个4环(girth 4),在置信传播迭代译码算法下可获得良好的性能。数值计算结果表明,对于该构造方法得到的GF(6t+1)和GF(12t+1)量子LDPC码,比基于BIBD的其他构造方法所得到的量子码的码长更长、量子校验矩阵更加稀疏、性能也更加优越。     

On Type II codes over F4

Previously, Type II codes over F₄ have been introduced as Euclidean self-dual codes with the property that all Lee weights are divisible by four. In this paper, a number of properties of Type II codes are presented. We construct several extremal Type II codes and a number of extremal Type I codes. It is also shown that there are seven Type II codes of length 12, up to permutation equivalence     

A systematic construction of self-dual codes

A new coding construction scheme of block codes using short base codes and permutations that enables the construction of binary self-dual codes is presented in Cadic et al. (2001) and Carlach et al. (1999, 2000). The scheme leads to doubly-even (resp,. singly-even) self-dual codes provided the base code is a doubly-even self-dual code and the number of permutations is even (resp., odd). We study the particular case where the base code is the [8, 4, 4] extended Hamming. In this special case, we construct a new [88, 44, 16] extremal doubly-even self-dual code and we give a new unified construction of the five [32, 16, 8] extremal doubly-even self-dual codes.     

关于q元Alternant码的最小距离

本文给出了ｑ元Ａｌｔｅｒｎａｎｔ码的最小距离一个新的下界，改进了以往有关结果。此外，本文将指出，文献［１］关于Ａｌｔｅｒｎａｎｔ码的最小距离的结果，一般是不成立的。     

A note on extended quadratic residue codes overGF(9)and their ternary images (Corresp.)

We discuss[2(p + 1), p + 1]double circulant codes which are the ternary images of the[p + 1,(p + 1)/2]extended quadratic residue codes over GF(9). Herepis a prime of the formp = 12k pm 5. As a special result we obtain a[64, 32,18]ternary self-dual code which is the largest known code meeting the bound of Mallows and Sloane.     

环F2+uF2上线性码及其对偶码的二元象

利用环F2＋uF2上线性码C的生成矩阵给出了码C的对偶码C^┴及其Gray象Ф（C）的生成矩阵，证明了环F2＋uF2上线性码及其对偶码的Gray象仍是对偶码。并由此给出了一个环F2＋uF2如上线性码为自对偶码的充要条件。