Cyclic Codes over the Integers Modulop

The purpose of this paper is twofold. First, we generalize the results of Pless and Qian and those of Pless, Solé, and Qian for cyclic ₄-codes to cyclic _p^m-codes. Second, we establish connections between this new development and the results on cyclic _p^m-codes obtained by Calderbank and Sloane. We produce generators for the cyclic _p^m-codes which are analogs to those for cyclic ₄-codes. We show that these may be used to produce a single generator for such codes. In particular, this proves that the ringR_n= _p^m[x]/(xⁿ− 1) is principal, a result that had been previously announced with an incorrect proof. Generators for dual codes of cyclic _p^m-codes are produced from the generators of the corresponding cyclic _p^m-codes. In addition, we also obtain generators for the cyclicp^m-ary codes induced from the idempotent generators for cyclicp-ary codes.     

On the Construction of Some Type II Codes over Z 4×Z4

We consider here the construction of Type II codes over the abelian group Z₄×Z₄. The definition of Type II codes here is based on the definitions introduced by Bannai [2]. The emphasis is given on the construction of these types of codes over the abelian group Z₄×Z₄ and in particular, the methods applied by Gaborit [7] in the construction of codes over Z₄ was extended to four different dualities with their corresponding weight functions (maps assigning weights to the alphabets of the code). In order to do this, we use the flattened form of the codes and construct binary codes analogous to the ones applied to Z₄ codes. Since each duality generates more than one weight function, we focus on those weights satisfying the squareness property. Here, by the squareness property, we mean that the weight function wt assigns the weight 0 to the Z₄×Z₄ elements (0, 0),(2, 2) and the weight 4 to the elements (0, 2) and (2, 0). The main results of this paper are focused on the characterization of these codes and provide a method of construction that can be applied in the generation of such codes whose weight functions satisfy the squareness property.     

A Criterion for Designs in ℤ4-codes on the Symmetrized Weight Enumerator

The Assmus–Mattson theorem is known as a method to find designs in linear codes over a finite field. It is an interesting problem to find an analog of the theorem for Z ₄-codes. In a previous paper, the author gave a candidate of the theorem. The purpose of this paper is to give an improvement of the theorem. It is known that the lifted Golay code over Z ₄ contains 5-designs on Lee compositions. The improved method can find some of those without computational difficulty and without the help of a computer.     

On (p a , p b , p a , p a-b )-Relative Difference Sets

This paper provides new exponent and rank conditions for the existence of abelian relative (p ^a,p ^b,p ^a,p ^a–b)-difference sets. It is also shown that no splitting relative (2^2c,2^d,2^2c,2^2c–d)-difference set exists if d > c and the forbidden subgroup is abelian. Furthermore, abelian relative (16, 4, 16, 4)-difference sets are studied in detail; in particular, it is shown that a relative (16, 4, 16, 4)-difference set in an abelian group G Z₈ × Z₄ × Z₂ exists if and only if exp(G) 4 or G = Z₈ × (Z₂)³ with N Z₂ × Z₂.     

Cyclic codes over the rings Z 2?+ uZ 2 and Z 2?+ uZ 2?+ u 2 Z 2

In this paper, we study cyclic codes over the rings Z ₂ + uZ ₂ and Z ₂ + uZ ₂ + u ² Z ₂ . We find a set of generators for these codes. The rank, the dual, and the Hamming distance of these codes are studied as well. Examples of cyclic codes of various lengths are also studied.      

Perfect codes in the graphs Ok

In this paper we consider the existence of perfect codes in the infinite class of distance-transitive graphs O_k. Perfect 1-codes correspond to certain Steiner systems and necessary conditions for the existence of such a code are satisfied if k + 1 is prime. We give some nonexistence results for perfect 2-, 3-, and 4-codes and for perfect e-codes in general, including a lower bound for k in terms of e.     

Efficient p-adic Cell Decompositions for Univariate Polynomials

Cell decompositions are constructed for polynomials f(x)Z_p[x] of degree n, such that n<p, using O(n²) cells. When f is square-free this yields a polynomial-time algorithm for counting and approximating roots in Z_p. These results extend to give a polynomial-time algorithm in the bit model for fZ[x].     

$${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -linear codes: generator matrices and duality

A code C{{\mathcal C}} is \mathbb Z₂\mathbb Z₄{{{\mathbb Z}_2}{{\mathbb Z}_4}} -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). In this paper \mathbb Z₂\mathbb Z₄{{{\mathbb Z}_2}{{\mathbb Z}_4}} -additive codes are studied. Their corresponding binary images, via the Gray map, are \mathbb Z₂\mathbb Z₄{{{\mathbb Z}_2}{{\mathbb Z}_4}} -linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity-check matrices are given. In order to do this, the appropriate concept of duality for \mathbb Z₂\mathbb Z₄{{{\mathbb Z}_2}{{\mathbb Z}_4}} -additive codes is defined and the parameters of their dual codes are computed.     

Codes over Zm

In this paper we study cyclic codes inZ _m. i.e., ideals inZ _mG, G a finite abelian group, and we give a classification of such codes.We also study the minimum Hamming distance and the generalized Hamming weight of BCH codes overZ _m.     

Codes with the Same Weight Distributions as the Goethals Codes and the Delsarte-Goethals Codes

The Goethals code is a binary nonlinear code of length 2^m+1 which has codewords and minimum Hamming distance 8 for any odd . Recently, Hammons et. al. showed that codes with the same weight distribution can be obtained via the Gray map from a linear code over Z ₄of length 2^m and Lee distance 8. The Gray map of the dual of the corresponding Z ₄ code is a Delsarte-Goethals code. We construct codes over Z ₄ such that their Gray maps lead to codes with the same weight distribution as the Goethals codes and the Delsarte-Goethals codes.     

The complexity of minimum difference cover

The complexity of searching minimum difference covers, both in Z⁺ and in Z_n, is studied. We prove that these two optimization problems are NP-hard. To obtain this result, we characterize those sets—called extrema—having themselves plus zero as minimum difference cover. Such a combinatorial characterization enables us to show that testing whether sets are not extrema, both in Z⁺ and in Z_n, is NP-complete. However, for these two decision problems we exhibit pseudo-polynomial time algorithms.     

An Assmus–Mattson-Type Approach for Identifying 3-Designs from Linear Codes over Z 4

The complete weight enumerator of the Delsarte–Goethals code over Z ₄ is derived and an Assmus–Mattson-type approach at identifying t-designs in linear codes over Z ₄ is presented. The Assmus–Mattson-type approach, in conjunction with the complete weight enumerator are together used to show that the codewords of constant Hamming weight in both the Goethals code over Z ₄ as well as the Delsarte–Goethals code over Z ₄ yield 3-designs, possibly with repeated blocks.     

On theG-matrices with entries and eigenvalues inQ(i)

LetG be a finite abelian group,K a subfield ofC, C[G] regarded as an algebra of matrices.A _G ^K {AC[G]| all the entries and eigenvalues ofA are inK} is an association algebra overK. In this paper, the association scheme ofA _G ^K is determined and in the caseK=Q(i), the first eigenmatrix of the association scheme computed. As an application, it is proved thatZ ₄×Z ₄×Z ₄ is the only abelian group admitted as a Singer group by some distance-regular digraph of girth 4 on 64 vertices.     

Combinatorial Bounds on Authentication Codes with Arbitration

Unconditionallysecure authentication codes with arbitration ( A²-codes)protect against deceptions from the transmitter and the receiveras well as that from the opponent. In this paper, we presentcombinatorial lower bounds on the cheating probabilities andthe sizes of keys of A²-codes. These bounds areall tight. Our main technique is a reduction of an A²-codeto a splitting A-code.     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z₊^d (d2)—the positive d-dimensional lattice points with partial ordering , let {X_k,kZ₊^d} be i.i.d. random variables with mean 0, and set S_n=∑_knX_k, nZ₊^d. We establish precise asymptotics for ∑_n|n|^r/p−2P(|S_n||n|^1/p), and for

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, (0δ1) as 0, and for

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as .     

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.     

Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions

A function f(x) defined on = ₁ × ₂ × … × _n where each _i is totally ordered satisfying f(x y) f(x y) ≥ f(x) f(y), where the lattice operations and refer to the usual ordering on , is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies −DΣ⁻¹D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.