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．     

Codes Over the p-adic Integers

We study codes over the p-adic integers and correct errors in the existing literature. We show that MDS codes exist over the p-adics for all lengths, ranks and p. We show that self-dual codes exist over the 2-adics if and only if the length is a multiple of 8 and that self-dual codes exist over the p-adics with p odd if and only if the length is 0 (mod 4) for p ≡ 3 (mod 4) and 0 (mod 2) for p ≡ 1 (mod 4).     

On Intersection Problem for Perfect Binary Codes

The main result is that to any even integer q in the interval 0 ≤ q ≤ 2^{n+1-2log(n+1)}, there are two perfect codes C₁ and C₂ of length n = 2^m − 1, m ≥ 4, such that |C₁ ∩ C₂| = q.     

Bounds on Codes Derived by Counting Components in Varshamov Graphs

We are interested in improving the Varshamov bound for finite values of length n and minimum distance d. We employ a counting lemma to this end which we find particularly useful in relation to Varshamov graphs. Since a Varshamov graph consists of components corresponding to low weight vectors in the cosets of a code it is a useful tool when trying to improve the estimates involved in the Varshamov bound. We consider how the graph can be iteratively constructed and using our observations are able to achieve a reduction in the over-counting which occurs. This tightens the lower bound for any choice of parameters n, k, d or q and is not dependent on information such as the weight distribution of a code. This work is taken from the author’s thesis [10]     

Two-point AG codes from the Beelen-Montanucci maximal curve

In this paper we investigate two-point algebraic-geometry codes (AG codes) coming from the Beelen-Montanucci (BM) maximal curve. We study properties of certain two-point Weierstrass semigroups of the curve and use them for determining a lower bound on the minimum distance of such codes. AG codes with better parameters with respect to comparable two-point codes from the Garcia-Güneri-Stichtenoth (GGS) curve are discovered.     

Two new bounds on the size of binary codes with a minimum distance of three

We prove two new upper bounds on the size of binary codes with a minimum distance of three, namelyA(10, 3)76 andA(11, 3)152.     

Spectral-Null Codes and Null Spaces of Hadamard Submatrices

Codes of length 2 ^m over {1, -1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of all have an rth order spectral null at zero frequency. Establishing the connection between and the parity-check matrix of Reed-Muller codes, the minimum distance of is obtained along with upper bounds on the redundancy of . An efficient algorithm is presented for encoding unconstrained binary sequences into .     

Nonbinary convolutional codes derived from group character codes

New families of unit memory as well as multi-memory nonbinary convolutional codes are constructed algebraically in this paper. These convolutional codes are derived from the class of group character codes. The proposed codes have basic generator matrices, consequently, they are noncatastrophic. Additionally, the new code parameters are better than the ones available in the literature.     

Important unanswered questions—1970

This paper was given as the opening address at the 1970 Albany International Conference on Radiation Effects in Semiconductors, and it attempts to establish a general overview of the field by concentrating on recent research developments and important unanswered questions. The continuing importance of impurity-defect interactions, of microscopic defect identification, and of the necessity for more theoretical calculations are emphasized. The rapid development of the field of ion implantation and its close relationship with radiation effects studies are pointed out. It is predicted that research in compound semiconductors will increase rapidly with close beneficial interaction with ion implantation studies.     

Projective Reed–Muller type codes on rational normal scrolls

In this paper we study an instance of projective Reed–Muller type codes, i.e., codes obtained by the evaluation of homogeneous polynomials of a fixed degree in the points of a projective variety. In our case the variety is an important example of a determinantal variety, namely the projective surface known as rational normal scroll, defined over a finite field, which is the basic underlining algebraic structure of this work. We determine the dimension and a lower bound for the minimum distance of the codes, and in many cases we also find the exact value of the minimum distance. To obtain the results we use some methods from Gröbner bases theory.