Self-dual group codes

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enumerators of linear codes over the ring of integers modulo 4. We also look at Gray maps from Galois rings to the finite field of prime size. We first give an application of the distance preserving Gray maps to obtain good ternary codes, and then we give an inductive algebraic construction of a distance preserving Gray map from Galois rings with the homogeneous distance to the field of prime size with the Hamming distance as well as a combinatorial construction. By using the Gray maps, we obtain some results about the weight enumerators of linear codes over the ring of dual numbers. In the last part of this work, we consider the permutation invariance of binary codes and the connection with linearity over certain rings. Among the rings considered, we have different variations and extensions of the ring of dual numbers. In this context we consider the Reed-Muller codes and answer questions about permutation invariance of Reed-Muller codes under certain permutation groups and the linearity of Reed-Muller codes over these aforementioned rings.

The ranks of cyclic and negacyclic codes over the finite chain ring R as well as their minimal generating sets are defined, and then the expression forms we presented by studying the structures of cyclic and negacyclic codes over the finite chain ring R. Through the paper, it is assumed that the length of codes n can not be divided by the characteristic of R.

On the ℓ-DLIPs of codes over finite commutative rings

Let p be a prime number and m be a positive integer. In this paper, we investigate cyclic codes of length n over the local non-Frobenius ring R=GR(p2,m)[u], where u2=0 and pu=0. We first determine the algebraic structure of cyclic codes of arbitrary length n. For the case gcd(n,p)=1, we explicitly describe the generators of cyclic codes over R. Moreover, we establish necessary and sufficient conditions for the existence of self-dual and LCD codes, together with their enumeration. Several illustrative examples and tables are presented, highlighting the mass formula for cyclic self-orthogonal codes, cyclic LCD codes, and families of new cyclic codes that arise from our results.

In this paper, we study constacyclic codes over finite principal ideal rings. An isomorphism between constacyclic codes and cyclic codes over finite principal ideal rings is given. Further, an open question is partially answered by giving necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite principal ideal rings. As an example of codes over a finite principal ideal ring, we study constacyclic codes over R+vR where v2=v and R is a finite chain ring.

In this chapter, we shall present some fundamental study on relatively self-dual codes over a finite commutative ring. Section 1.1 is devoted to the basic definitions and properties of such codes. In Section 1.2, we shall present our gluing technique for constructing relatively self-dual codes. Several exotic general constructions of self-dual codes over ℤn will be given in Section 1.3. Four constructions of binary self-dual codes by our gluing technique will be given in Section 1.4. In Section 1.5, we shall give some constructional examples of ternary self-dual codes. In Section 1.6, we shall present some families of self-dual codes over ℤ4. Sections 1.1, 1.2 and Constructions III, IV in Section 1.4 were published in [X2].

We give a 1-level squaring construction for binary repeated-root cyclic codes of length n=2/sup a/b, a/spl ges/1, b odd. This allows us to obtain the weight distributions of all cyclic binary self-dual codes of lengths up to 110, which are not accessible by direct computation. We also use the shadow construction, as a particular method for Type I self-dual codes.

Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code C, identify for each dimension v, the smallest size of the support of a sub code of $C$ of dimension u, The GHW of a code are of interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use in bounding the state complexity of the trellis representation of the code. In prior work [1] by the same authors, a code-shortening algorithm was employed to derive upper bounds on the GHW of binary projective, Reed-Muller (PRM) codes. In the present paper, we derive a matching lower bound by adapting the proof techniques used originally for Reed-Muller (RM) codes by Wei in [2]. This results in a characterization of the GHW hierarchy of binary PRM codes.

Cyclic [formula omitted]-additive codes

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p e , l) (including $${\mathbb{Z}_{p^e}}$$ ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 e , l) of length n = 2 l for any a ? 1 and l ? 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.

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We compute the covering radius of each binary cyclic code of length < 64 (for both even and odd lengths) and redundancy < 28 .We also compute the covering radii of their punctured codes and shortened codes.Thus we give exact covering radii of over six thousand codes.For each of these codes (except for certain composite codes), we also determine the number of cosets of each weight less than or equal to the covering radius.These results are used to compute the minimum distances of the above cyclic codes.We use the covering radii of shortened codes and other criteria for normality to show that all but eight of the cyclic codes for which we determine the covering radius are normal.For all but seven of these normal codes, we determine the norm using some old results and some new results proved here.We observe that many cyclic codes are among the best covering codes discovered so far, and some of them lead to improvements on the previously published bounds on t[n , k], the smallest covering radius of any binary linear [n , k] code.Among some other applications of our results, we use our table of covering radii and a code augmentation argument to give four improvements on the values of dmax(n , k) .wheredmix{n, k) is the largest minimum distance of any binary [n , k] code.These results show that the covering radius is intimately connected with the other three parameters of a linear code, n, k , and d .We also give a complete classification (up to isomorphism) of cyclic self-dual codes of lengths 42, 56, and 60.The computations were carried out mainly on concurrent machines (hypercubes and Connection Machines); we give a description of our algorithm.

Repeated root Cyclic and Negacyclic codes over Galois rings havebeen studied much less than their simple root counterparts. This situation isbeginning to change. For example, repeated root codes of length ps, where $p$ isthe characteristic of the alphabet ring, have been studied under some additionalhypotheses. In each one of those cases, the ambient space for the codes hasturned out to be a chain ring. In this paper, all remaining cases of cyclicand negacyclic codes of length ps over a Galois ring alphabet are considered.In these cases the ambient space is a local ring with simple socle but not achain ring. Nonetheless, by reducing the problem to one dealing with uniserialsubambients, a method for computing the Hamming distance of these codes isprovided.

Using representation theoretical methods we investigate self-dual group codes and their extensions in characteristic 2. We prove that the existence of a self-dual extended group code heavily depends on a particular structure of the group algebra KG which can be checked by an easy-to-handle criteria in elementary number theory. Surprisingly, in the binary case such a code is doubly even if the converse of Gleason's theorem holds true, i.e., the length of the code is divisible by 8. Furthermore, we give a short representation theoretical proof of an earlier result of Sloane and Thompson which states that a binary self-dual group code is never doubly even if the Sylow 2-subgroups of G are cyclic. It turns out that exactly in the case of a cyclic or Klein four group as Sylow 2-subgroup doubly even group codes do not exist.

New metrics and distances for linear codes over the ring Fq[u]/(u) are defined, which generalize the Gray map, Lee weight, and Bachoc weight; and new bounds on distances are given. Two characterizations of self-dual codes over Fq[u]/(u) are determined in terms of linear codes over Fq. An algorithm to produce such self-dual codes is also established. 1.Introduction. Many optimal codes have been obtained by studying codes over general rings rather than fields. Lately, codes over finite chain rings (of which Fq[u]/(u) is an example) have been a source of many interesting properties [6, 8, 3]. Gulliver and Harada [4] found good examples of ternary codes over F3 using a particular type of Gray map. Siap and Ray-Chaudhuri in [9] established a relation between codes over Fq[u]/(u − a) and codes over Fq, which was used to obtained new codes over F3 and F5. In this paper we present a certain generalization of the method used in [4] and [9], defining a family of metrics for linear codes over Fq[u]/(u) and obtaining as particular examples the Gray map, the Gray weight, the Lee weight and the Bachoc weight. For the latter, we give a new bound on the distance of those codes. It also shows that the Gray images of codes over F2 + uF2 are more powerful than codes obtained by the so-called u—u+v condition. With these tools in hand, we study conditions for self-duality of codes over Fq[u]/(u). In [7] the authors study the case of self-dual cyclic codes in terms of the generator polynomials. In this paper we study self-dual codes in terms of linear codes over Fq that are obtained as images under the maps defined on the first part of the paper. We provide a way to construct many self-dual codes over Fq starting from a self-dual code over Fq[u]/(u). We also study self-dual codes in terms of the torsion codes, and provide a way to construct many self-dual codes over Fq[u]/(u) starting from a self-orthogonal code over Fq. Our results contain many of the properties studied by Bachoc for self-dual codes over F3 + uF3 in [1]. 2. Metric for Codes over Fq[u]/(u). We will use R(q, t) to denote the commutative ring Fq[u]/(u). The q elements of this ring can be represented in two different forms, and we will use the most appropriate in each case. First, we can use the ∗The project was partially supported by Office of Research of the University of Michigan-Flint. †2000 Mathematics Subject Classification: Primary: 94B05, 94B60, Secondary: 11T71. ‡