Symmetry codes over GF(3)

A lower bound $\text{{}\text{1}\text{4}\text{(4q + 5)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+}\text{3}\text{2}$ is given for the minimum weight of the symmetry code C(q) over GF(3), which is introduced by Pless [3].     

On the classification of all self-dual additive codes over GF(4) of length up to 12

We consider additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these codes can be represented as graphs, and that two codes are equivalent if and only if the corresponding graphs are equivalent with respect to local complementation and graph isomorphism. We use these facts to classify all codes of length up to 12, where previously only all codes of length up to 9 were known. We also classify all extremal Type II codes of length 14. Finally, we find that the smallest Type I and Type II codes with trivial automorphism group have length 9 and 12, respectively.     

New linear codes over GF(8)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d₈(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 8]-code for given values of n and k. In this paper, eighteen new linear codes over GF(8) are constructed which improve the table of d₈(n, k) by Brouwer.     

The weight distribution of linear codes over GF(ql) having generator matrix over GF(q>)

We study the weight distribution of the linear codes over GF(q^l) which have generator matrices over GF(q) and their dual codes. As an application we find the weight distribution of the irreducible cyclic (23(2¹≈1), 111) codes over GF(2) for all lnot divisible by 11.     

Linear cyclic codes of wordlength v over GF(q s ) which are also linear cyclic codes of wordlength sv over GF(q)

Any nonsingular linear transformation : GF(q^s) GF(q^s) can be used to treat a linear cyclic code of wordlength v over GF(q^s) as a linear code () of Wordlength sv over GF(q). This paper determines those linear cyclic codes and transformations for which the resulting linear code () is also cyclic.     

利用有限域GF（2m）上的三项式构造二元循环码

循环码是一类特殊的线性码,由于循环码快速的编码和译码算法,它被广泛应用于消费电子,数据存储以及通信系统当中.在本文中,利用特征是偶数的有限域上的三项式构造出了两类二元循环码,我们不仅可以确定出这两类循环码最小距离的下界,而且这两类循环码在参数的选取上非常的灵活.     

On the matroids representable over GF(4)

The purpose of this note is to present a counterexample to a conjecture of Kahn and Seymour on the minor-minimal matroids not representable over GF(4).     

Designs in Additive Codes over GF(4)

In this paper we discuss the security of digital signature schemes based on error-correcting codes. Several attacks to the Xinmei scheme are surveyed, and some reasons given to explain why the Xinmei scheme failed, such as the linearity of the signature and the redundancy of public keys. Another weakness is found in the Alabbadi-Wicker scheme, which results in a universal forgery attack against it. This attack shows that the Alabbadi-Wicker scheme fails to implement the necessary property of a digital signature scheme: it is infeasible to find a false signature algorithm D from the public verification algorithm E such that E(D ( )) = for all messages . Further analysis shows that this new weakness also applies to the Xinmei scheme.     

Construction of pseudorandom binary sequences using additive characters over GF(2 k )

In a series of papers Mauduit and Sárközy (partly with coauthors) studied finite pseudorandom binary sequences and they constructed sequences with strong pseudorandom properties. In these constructions fields with prime order were used. In this paper a new construction is presented, which is based on finite fields of order 2 ^k .     

Constructing differentially 4-uniform permutations over GF(22m ) from quadratic APN permutations over GF(22m+1)

In this paper, by means of the idea proposed by Carlet (ACISP 1-15, 2011), differentially 4-uniform permutations with the best known nonlinearity over \({\mathbb{F}_{2^{2m}}}\) are constructed using quadratic APN permutations over \({\mathbb{F}_{2^{2m+1}}}\) . Special constructions are given using the Gold functions. The algebraic degree of the constructions and their compositional inverses is also investigated. One construction and its compositional inverse both have algebraic degree m + 1 over \({\mathbb{F}_2^{2m}}\) .     

Self-dual codes over GF(4)

This paper studies codes $\text{C}$ over GF(4) which have even weights and have the same weight distribution as the dual code C^⊥. Some of the results are as follows. All such codes satisfy $\text{C}{}^{\mathrm{\perp }}\text{=}\text{C}$ (If C^⊥= C, $\text{T}$ has a binary basis.) The number of such C's is determined, and those of length ?14 are completely classified. The weight enumerator of $\text{C}$ is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for $\text{C}$ to be extended cyclic. Two new 5-designs are constructed. A generator matrix for $\text{C}$ can be taken to have the form [I | B], where $\text{B}{}^{\mathrm{\perp }}\text{=}\text{B}$. We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.     

Binary images of some self-dual codes over GF(2m) with respect to trace-orthogonal basis

The note deals with a class of self-dual binary codes containing external doubly-even codes for lengths 8, 16, 24, 32, 40 and 64. For most of them, our construction also provides a simple method for calculating the distance of the code. To our knowledge, this is the first example of a (64, 32, 12) extermal doubly-even self-dual code.     

Tangential 1-blocks over GF(3)

In a fundamental paper on the algebraic theory of graph colouring Tutte [2] proves that the only tangential 1-block over GF(2) is the geometry corresponding to the polygon matroid of the graph K₃. For a good discussion of the problem and a direct proof of Tutte's theorem see Aigner [1, p. 373].

The purpose of this note is to prove the following result.     

Latroids and their representation by codes over modules

It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code is an evaluation of the Tutte polynomial of the matroid, and a standard identity relating the Tutte polynomials of dual matroids gives rise to a MacWilliams identity relating the weight enumerators of dual codes. More recently, codes over rings and modules have been considered, and MacWilliams type identities have been found in certain cases.

In this paper we consider codes over rings and modules with code duality based on a Morita duality of categories of modules. To these we associate latroids, defined here. We generalize notions of deletion, contraction, minors and duality, on both codes and latroids, and examine all natural relations among these.

We define generating functions associated with codes and latroids, and prove identities relating them, generalizing above-mentioned generating functions and identities.

     

Inhomogenous quantum codes (I): additive case

In this paper, the quantum error-correcting codes are generalized to the inhomogenous quantum-state space $ \mathbb{C}^{q_1 } \otimes \mathbb{C}^{q_2 } \otimes \cdots \otimes \mathbb{C}^{q_n } $ , where q _i (1 ? i ? n) are arbitrary positive integers. By attaching an abelian group A _i of order q _i to the space Cqi $ \mathbb{C}^{q_1 } \left( {1 \leqslant i \leqslant n} \right) $ , we present the stabilizer construction of such inhomogenous quantum codes, called additive quantum codes, in term of the character theory of the abelian group A = A ₁⊕A ₂⊕...⊕? _n . As usual case, such construction opens a way to get inhomogenous quantum codes from the classical mixed linear codes. We also present Singleton bound for inhomogenous additive quantum codes and show several quantum codes to meet such bound by using classical mixed algebraic-geometric codes.