Negacyclic duadic codes

This paper extends the concepts from cyclic duadic codes to negacyclic codes over F_q (q an odd prime power) of oddly even length. Generalizations of defining sets, multipliers, splittings, even-like and odd-like codes are given. Necessary and sufficient conditions are given for the existence of self-dual negacyclic codes over F_q and the existence of splittings of 2N, where N is odd. Other negacyclic codes can be extended by two coordinates in a way to create self-dual codes with familiar parameters.     

The weight distributions of irreducible cyclic codes of length

Let m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2^m over F_q are determined explicitly.     

Enumeration of inequivalent irreducible Goppa codes

We consider irreducible Goppa codes over F_q of length qⁿ defined by polynomials of degree r, where q is a prime power and n,r are arbitrary positive integers. We obtain an upper bound on the number of such codes.     

A symmetric Roos bound for linear codes

The van Lint-Wilson AB-method yields a short proof of the Roos bound for the minimum distance of a cyclic code. We use the AB-method to obtain a different bound for the weights of a linear code. In contrast to the Roos bound, the role of the codes A and B in our bound is symmetric. We use the bound to prove the actual minimum distance for a class of dual BCH codes of length q²−1 over Fq. We give cyclic codes [63,38,16] and [65,40,16] over F₈ that are better than the known [63,38,15] and [65,40,15] codes.     

The lengths of Hermitian self-dual extended duadic codes

Duadic codes are a class of cyclic codes that generalize quadratic residue codes from prime to composite lengths. For every prime power q, we characterize integers n such that there is a duadic code of length n over F_q² with a Hermitian self-dual parity-check extension. We derive asymptotic estimates for the number of such n as well as for the number of lengths for which duadic codes exist.     

环F3+vF3上的循环码与常循环码

本文研究了环F₃+vF₃上的循环码与常循环码.通过环F₃+vF₃与域F₃上的循环码之间关系,证明了环F₃+vF₃上循环码是由一个多项式生成的.最后,用类似的方法,得到了环F₃+vF₃上v-常循环码也是由一个多项式生成的.     

Constructing quasi-cyclic codes from linear algebra theory

Let F _q be a finite field of cardinality q, l and m be positive integers and M _l (F _q ) the F _q -algebra of all l × l matrices over F _q . We investigate the relationship between monic factors of X ^m ? 1 in the polynomial ring M _l (F _q )[X] and quasi-cyclic (QC) codes of length lm and index l over F _q . Then we consider the idea of constructing QC codes from monic factors of X ^m ? 1 in polynomial rings over F _q -subalgebras of M _l (F _q ). This idea includes ideas of constructing QC codes of length lm and index l over F _q from cyclic codes of length m over a finite field F _q l, the finite chain ring F _q  + uF _q  + · · · + u ^{l ? 1} F _q (u ^l  = 0) and other type of finite chain rings.     

Some constructions of linearly optimal group codes

We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n,n-3,3]_q-codes over F_q for n=2q and n=3q. These codes are linearly optimal, i.e. have maximal dimension among linear codes having a given length and distance. Although codes with such parameters are known, our main results state that we can construct such codes as (left) group codes. In the paper we use a construction of Reed-Solomon codes as ideals of the group ring F_qG where G is an elementary abelian group of order q.     

On some special codes over Fp + uFp + uFp

We define a new map between codes over F_p + uF_p + u²F_p and F_p which is different to that defined in [2]. It is proved that the image of the linear cyclic code over the commutative ring F_p + uF_p + u²F_p with length n under this map is a distance-invariant quasi-cyclic code of index p² with length p²n over F_p. Moreover, it is proved that, if (n, p) = 1, then every code with length p²n over F_p which is the image of a linear (1 − u²)-cyclic code with length n over F_p + uF_p + u²F_p under this map is permutation equivalent to a quasi-cyclic code of index p².     

On self-orthogonal group ring codes

We obtain structural results about group ring codes over F[G], where F is a finite field of characteristic p > 0 and the Sylow p-subgroup of the Abelian group G is cyclic. As a special case, we characterize cyclic codes over finite fields in the case the length of the code is divisible by the characteristic of the field. By the same approach we study cyclic codes of length m over the ring R = F _q [u], u ^r  = 0 with r  > 0, gcd(m, q) = 1. Finally, we give a construction of quasi-cyclic codes over finite fields.      

Intersections of q-ary perfect codes

The intersections of q-ary perfect codes are under study. We prove that there exist two q-ary perfect codes C ₁ and C ₂ of length N = qn + 1 such that |C ₁ ? C ₂| = k · |P _i |/p for each k ∈ {0,..., p · K ? 2, p · K}, where q = p ^r , p is prime, r ≥ 1, $n = \tfrac{{q^{m - 1} - 1}}{{q - 1}}$ , m ≥ 2, |P _i | = p ^nr(q?2)+n , and K = p ^{n(2r?1)?r(m?1)}. We show also that there exist two q-ary perfect codes of length N which are intersected by p ^nr(q?3)+n codewords.     

Minimal quadratic residue cyclic codes of length 2 n

LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 ⁿ (n>1) be ?(2 ⁿ )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m?3, then the ring $$R_{2^n } = F\left[ x \right]/< x^{2^n } - 1 > $$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length 2 ⁿ generated by these idempotents are completely described. Ifq=8m+3 then the expressions for the 2n?1 primitive idempotents ofR ₂ ⁿ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n?1 idempotents are also obtained. The casen=2, 3 is dealt separately.     

The nonexistence of ternary [105,6,68] and [230,6,152] codes

Let [n,k,d]_q-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). In this paper, the nonexistence of [105,6,68]₃ and [230,6,152]₃ codes is proved.     

Direct constructions of additive codes

A code is q^m‐ary q‐linear if its alphabet forms an m‐dimensional vector space over ??_q and the code is linear over ??_q. These additive codes form a natural generalization of linear codes. Our main results are direct constructions of certain families of additive codes. These comprise the additive generalization of the Kasami codes, an additive generalization of the Bose‐Bush construction of orthogonal arrays of strength 2 as well as a class of additive codes which are being used for deep space communication. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 207–216, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.20000     

关于环Fpm+uFpm+u2Fpm上循环码的注记

本文研究了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码分类.通过建立环F_p^m+uF_p^m+u²F_p^m到环F_p^m+uF_p^m的同态,给出了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码的新分类方法.应用这种方法,得到了环F_p^m+uF_p^m+u²F_p^m长度为p^s的循环码的码词数.     

Three-dimensional cyclic Fire codes

Three-dimensional cyclic array codes over F _q that can correct single three-dimensional bursts (or clusters) of errors are considered. The class cyclic three-dimensional burst-error-correcting array codes, called three-dimensional Fire codes, is constructed. Several important properties such as the burst-error-correcting capability and the positions of the parity-check symbols are presented. Also, encoding and decoding algorithms are given.     

Fq-linear Cyclic Codes over $$ F{_q^m}$$ : D FT Approach

Codes over that are closed under addition, and multiplication with elements from F_q are called F_q-linear codes over . For m 1, this class of codes is a subclass of nonlinear codes. Among F_q-linear codes, we consider only cyclic codes and call them F_q-linear cyclic codes (F_q LC codes) over The class of F_q LC codes includes as special cases (i) group cyclic codes over elementary abelian groups (q=p, a prime), (ii) subspace subcodes of Reed–Solomon codes (n=q^m–1) studied by Hattori, McEliece and Solomon, (iii) linear cyclic codes over F_q (m=1) and (iv) twisted BCH codes. Moreover, with respect to any particular F_q-basis of , any F_qLC code over can be viewed as an m-quasi-cyclic code of length mn over F_q. In this correspondence, we obtain transform domain characterization of F_q LC codes, using Discrete Fourier Transform (DFT) over an extension field of The characterization is in terms of any decomposition of the code into certain subcodes and linearized polynomials over . We show how one can use this transform domain characterization to obtain a minimum distance bound for the corresponding quasi-cyclic code. We also prove nonexistence of self dual F_q LC codes and self dual quasi-cyclic codes of certain parameters using the transform domain characterization.AMS classification 94B05     

The Galois variance of constacyclic codes

This paper studies the Galois images of constacyclic codes over ${\mathbb{F}}_{q^m}$ of length relatively prime to q, and determines when those images are equal and when they intersect only at the zero codeword. The subfield subcodes and trace codes of constacyclic codes are also determined.     

Some New Results on Optimal Codes Over F5

We present some results on almost maximum distance separable (AMDS) codes and Griesmer codes of dimension 4 over over the field of order 5. We prove that no AMDS code of length 13 and minimum distance 5 exists, and we give a classification of some AMDS codes. Moreover, we classify the projective strongly optimal Griesmer codes over F₅ of dimension 4 for some values of the minimum distance.