环F_p+uF_p+vF_p+uvF_p上的循环码

讨论了非有限链环R=F_p+uF_p+vF_p+uvF_p上的循环码.通过环R上的循环码与多项式环R_n=(F_p+uF_p+vF_p+uvF_p)[x]/(xⁿ-1)的理想的对应关系及对R_n的研究给出了R上循环码的刻画.最后定义了一个Gray映射,并刻画了F_p+uF_p+vF_p+uvF_p上的循环码在该映射下的像.     

Affine Invariant and Cyclic Codes over p-adic Numbers and Finite Rings

Affine invariant and cyclic codes over p-adic numbers and over integers modulo p ^d are studied. It has been determined what cyclic codes have an extension that is affine invariant.     

环F_3+uF_3+vF_3+uvF_3上的循环码

研究了环R=F_3+uF_3+vF_3+uvF_3上循环码的结构(u~2=u,v~2=v,uu=uu),证明了该环上的循环码是主理想生成的,并给出了其上循环码的生成多项式.     

有限链环上循环码与负循环码的生成多项式

从另一种角度研究了有限链环上循环码.给出了这种环上循环码的构造由这种构造得到了有限链环上的循环码的生成多项式.借助有限链环上循环码与负循环码的同构,也得到了这种环上循环码的生成元.     

环F_2+uF_2上长度为2n(n为奇数)的循环码

研究了环F2+uF2上长度为2n(n为奇数)的循环码,给出了循环码及其对偶码的生成多项式,以及循环码为自对偶码的充要条件,最后进一步给出了循环码极小Lee重量的一些相关结论     

Duals of constacyclic codes over finite local Frobenius non-chain rings of length 4

Duals of constacyclic codes over a finite local Frobenius non-chain ring of length 4, the length of which is relatively prime to the characteristic of the residue field of the ring are determined. Generators for the dual code are obtained from those of the original constacyclic code. In some cases self-dual codes are determined.     

Construction of Certain Cyclic Distance-Preserving Codes Having Linear-Algebraic Characteristics

An ordered list of binary words of length n is called a distance-preserving m, n-code, if the list distance between two words is equal to their Hamming distance, for distances up to m. A technique for constructing cyclic m, n-codes is presented, based on the standard Gray code and on some simple tools from linear algebra.     

The Covering Radius of Ternary Cyclic Codes with Length up to 25

The covering radius of all ternary cyclic codes of length up to 25 is given. Some of the results were obtained by computer and for others mathematical reasonings were applied. The minimal distances of all codes were recalculated.     

环F_2+uF_2上偶长的(1+u)-常循环码

给出了环F2+uF2上任意偶长的(1+u)-常循环码的结构,确定了给定偶长度F2+uF2上(1+u)-常循环码的数目.通过Gray映射,得到了F2+uF2上偶长的(1+u)-常循环码的二元象.     

Zpk+1环上的循环码的Gray像

定义了Znpk+1到Znpkp的Gray映射,给出该映射的一个性质,证明了Zpk+1环上码长为n的码为循环码的充要条件是它的Gray像是Zp上长度为npk指数为pk的准循环码.     

Cyclic Codes Over\mathbb{Z}_{4} of Even Length

We determine the structure of cyclic codes over for arbitrary even length giving the generator polynomial for these codes. We determine the number of cyclic codes for a given length. We describe the duals of the cyclic codes, describe the form of cyclic codes that are self-dual and give the number of these codes. We end by examining specific cases of cyclic codes, giving all cyclic self-dual codes of length less than or equal to 14. San Ling - The research of the second named author is partially supported by research Grants MOE-ARF R-146-000-029-112 and DSTA R-394-000-011-422.     

环Fq+uFq+vFq上的循环码和量子码

本文研究了环R=F_q+uF_q+vF_q(u²=u,v²=v,uv=vu=0)上的循环码构造量子码的方法.利用环R上循环码的分解与生成多项式,给出了R上一个循环码可以构造量子码的一个充要条件.作为这类循环码的应用,得到了新的非二元量子码.     

环F_(p~k)+uF_(p~k)+u~2F_(p~k)上的常循环码和循环码

记环R=F_(p~k)+uF_(p~k)+u~2F_(p~k),定义了一个从R~n到F_(p~k)~(2np~k)的Gray映射.利用Gray映射的性质,研究了环R上(1-u~2)-循环码和循环码.证明了环R上码是(1-u~2)-循环码当且仅当它的Gray象是F_(p~k)上的准循环码.当(n,p)=1时,证明了环R上的长为n的线性循环码的Gray象置换等价于域F_(p~k)上的线性准循环码.     

The Theory of Cyclic Codes and a Generalization to Additive Codes

We present a new approach to the theory of cyclic and constacyclic codes and generalize the theory to cover the family of additive (not necessarily linear) cyclic codes. The approach is based on the action of the Galois group (cyclotomic cosets). The conventional representation of cyclic codes as ideals in a factor ring of the polynomial ring is not needed.     

关于循环阵列码

本文用代数观点来研究循环阵列码,证明了一般的阵列码是一些极小循环阵列码的直和,并且对极小循环阵列码给出了明确的刻画．当有限域的特征不整除群的阶时,给出了直接写出相应的多项式环的本原幂等元的方法,从而可以直接写出所有的极小循环码．     

Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Let $R$ be the Galois ring $GR(p^k,s)$ of characteristic $p^k$ and cardinality $p^{sk}$. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length $n$ over $R$, and a $p$-adic integer ring with $gcd(p,n)=1$. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length $r{l}^m$ over $R$, where $r,l,$ and $t$ are three primes with $2?l$, $r|(q^t?1)$, $l^v\parallel (q^t?1)$ and $gcd(rl,q(q?1))=1$. Finally, as applications, weight distributions of all irreducible cyclic codes for $t=2$ and generator polynomials of self-dual cyclic codes of length $l^m$ and $r{l}^m$ over $R$ are given.     

环F_2+uF_2上长为2~e循环码的Gray象

利用环F2+uF2上长为2e的循环码结构,证明了这样的循环码的一类码在Gray映射下的象是循环码,并给出了环F2+uF2上长为2e的循环码的Gray象仍是循环码的一个充要条件.