A survey of constructive coding theory, and a table of binary codes of highest known rate (original) (raw)
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Algorithmic issues in coding theory
Lecture Notes in Computer Science, 1997
The goal of this article is to provide a gentle introduction to the basic definitions, goals and constructions in coding theory. In particular we focus on the algorithmic tasks tackled by the theory. W e describe some of the classical algebraic constructions of error-correcting codes including the Hamming code, the Hadamard code and the Reed Solomon code. We describe simple proofs of their error-correction properties. We also describe simple and e cient algorithms for decoding these codes. It is our aim that a computer scientist with just a basic knowledge of linear algebra and modern algebra should be able to understand every proof given here. We also describe some recent developments and some salient open problems.
SOME NEW RESULTS ON BINARY LINEAR BLOCK CODES
Certain properties of the parity-check matrix H of (n, k) linear codes are used to establish a computerised search procedure for new binary linear codes. Of the new error-correcting codes found by this procedure, two codes were capable of correcting up to two errors, three codes up to three errors, four codes up to four errors and one code up to five errors. Two meet the lower bound given by Helgert and Stinaff, and seven codes exceed it. In addition, one meets the upper bound. Of the even-Hamming-distance versions of these codes, eight meet the upper bound, and the remaining two exceed the lower bound.
IEEE Transactions on Information Theory, 1972
In this paper constructions are given for combining two, three, or four codes to obtain new codes. The AndryanovSaskovets construction is generalized. It is shown that the Preparata double-errorcorrecting codes may be extended by about (block length)"' symbols, of which only one is a check symbol, and that e-error-correcting BCH codes may sometimes be extended by (block length)"' symbols, of which only one is a check symbol. Several new families of linear and nonlinear double-error-correcting codes are obtained. Finally, an infinite family of linear codes is given with d/n = 3, the 8rst three being the (24,2",8) Golay code, a (48,215,16) code, and a (96,218,32) code. Most of the codes given have more codewords than any comparable code previously known to us. Dejinitions N (n,M,d) code %' is a set of M binary vectors of A length n, any two of which differ in at least d places.
A perspective on coding theory
Information Sciences, 1991
The field of error correcting codes has developed rapidly over the past forty years. During the past decade in particular two very significant developments have occurred and these are briefly reviewed here. From this basis, suggestions are made as to where coding principles might find application in the future.
Computational Hardness and Explicit Constructions of Error Correcting Codes
Fourty-Fourth Annual …, 2006
We outline a procedure for using pseudorandom generators to construct binary codes with good properties, assuming the existence of sufficiently hard functions. Specifically, we give a polynomial time algorithm, which for every integers n and k, constructs polynomially many linear codes of block length n and dimension k, most of which achieving the Gilbert-Varshamov bound. The success of the procedure relies on the assumption that the exponential time class of E def = DTIME[2 O(n) ] is not contained in the sub-exponential space class DSPACE[2 o(n) ].
On the algebraic representation of selected optimal non-linear binary codes
2012 IEEE International Symposium on Information Theory Proceedings, 2012
Revisiting an approach by Conway and Sloane we investigate a collection of optimal non-linear binary codes and represent them as (non-linear) codes over Z4. The Fourier transform will be used in order to analyze these codes, which leads to a new algebraic representation involving subgroups of the group of units in a certain ring. One of our results is a new representation of Best's (10, 40, 4) code as a coset of a subgroup in the group of invertible elements of the group ring Z4[Z5]. This yields a particularly simple algebraic decoding algorithm for this code. The technique at hand is further applied to analyze Julin's (12, 144, 4) code and the (12, 24, 12) Hadamard code. It can also be used in order to construct a (non-optimal) binary (14, 56, 6) code.