Vera Pless

Fundamentals of Error-Correcting Codes: Fundamentals of Error-Correcting Codes

Preface 1. Basic concepts of linear codes 2. Bounds on size of codes 3. Finite fields 4. Cyclic codes 5. BCH and Reed-Soloman codes 6. Duadic codes 7. Weight distributions 8. Designs 9. Self-dual codes 10. Some favourite self-dual codes 11. Covering radius and cosets 12. Codes over Z4 13. Codes from algebraic geometry 14. Convolutional codes 15. Soft decision and iterative decoding Bibliography Index.

Fundamentals of Error-Correcting Codes: Contents

Preface 1. Basic concepts of linear codes 2. Bounds on size of codes 3. Finite fields 4. Cyclic codes 5. BCH and Reed-Soloman codes 6. Duadic codes 7. Weight distributions 8. Designs 9. Self-dual codes 10. Some favourite self-dual codes 11. Covering radius and cosets 12. Codes over Z4 13. Codes from algebraic geometry 14. Convolutional codes 15. Soft decision and iterative decoding Bibliography Index.

Cyclic codes and quadratic residue codes over Z/sub 4/

A set of n-tuples over Z/sub 4/ is called a code over Z/sub 4/ or a Z/sub 4/ code if it is a Z/sub 4/ module. We prove that any Z/sub 4/-cyclic code C has generators of the form (fh, 2fg) where fgh=x/sup n/-1 over Z/sub 4/ and |C|=4/sup deg g/2/sup deg h/. We also show that C/sup /spl perp// has generators of the form (g/sup */h/sup */, 2f/sup */g/sup */). We show that idempotent generators exist for certain cyclic codes. A particularly interesting family of Z/sub 4/-cyclic codes are quadratic residue codes. We define such codes in terms of their idempotent generators and show that these codes also have many good properties which are analogous in many respects to properties of quadratic residue codes over a field. We show that the nonlinear binary images of the extended QR codes of lengths 32 and 48 have higher minimum weights than comparable known linear codes.

On the existence of a certain (64, 32, 12) extremal code

J.H. Conway and N.J.A. Sloane (see ibid., vol.36, no.6, p.1319-33, Nov. 1990) give weight enumerators of several self-dual codes with the highest possible minimal distance whose existence was not known. A generator matrix for one of these, a Type I (64, 32, 12) code, is given, proving its existence. The method of construction is described. >

Decoding binary R(2,5) by hand and by machine

We decode the binary Reed-Muller [32,16,8] code R(2,5) by hand by two methods. One, the representation decoding method, is the analogue of the method used to decode the Golay code. The other is the new syndrome decoding method. We also decode R(2,5) by machine using information sets.

Cyclic self-dual Z/sub 4/-codes

Binary cyclic self-dual codes of odd length do not exist. It is not the case, however, of Z/sub 4/-cyclic self-dual codes. Some of the first examples, aside from the trivial self-dual code, are supplemented quadratic residue codes. The aim of this paper is to characterize arithmetically the (odd) lengths where such non-trivial cyclic self-dual Z/sub 4/-codes can exist and to give some examples for short lengths. As the length is odd, it is hopeless to try to obtain directly Type II codes; i.e. codes whose Euclidean weights are multiples of 8 since these exist only for lengths a multiples of 8. It is possible, nonetheless, to obtain Type I Z/sub 4/-codes, and, accordingly, by construction A, Type I lattices. We obtain, in that way, the only two extremal odd lattices in dimensions 15-47: the shorter Leech lattice O/sub 23/ in dimension 23 and A/sub 15//sup +/, in dimension 15. Invariant theory enables us to compute the symmetric weight enumerators of the codes and therefore the theta series of the associated lattices, including the norm and kissing number thereof. In passing, we mention an amusing non-existence arithmetic criterion for cyclic projective planes.

Type II codes over F/sub 4/

The natural analogues of Lee weight and Gray map over F/sub 4/ are introduced. Self-dual codes for the Euclidean scalar product with Lee weights multiples of 4 are called Type II. They produce Type II binary codes by Gray map. All extended Q-codes of length multiples of 4 are Type II, this includes generalized quadratic residue codes attached to a prime power q/spl equiv/7 (mod 8). Certain double circulant codes are also considered. The first binary extremal singly-even [92,46,16] self-dual code is constructed. A general mass formula is derived.

All Z/sub 4/ codes of Type II and length 16 are known

Using a new computer algorithm that determines the automorphism group of a Z/sub 4/ code, and efficient ways to find generator matrices of Type II codes, we show that there are 133 inequivalent Type II Z/sub 4/ codes of length 16.

Parents, children, neighbors and the shadow [binary code theorems]

Discusses five code theorems. The author brings together several concepts with interesting relations to each other. All the codes are binary. The weights of all vectors in a self-orthogonal code must be even, however all weights in a code can be even without the code being self-orthogonal. The author calls C even if all its weights are even, and calls C doubly-even (d.e.), if the weights of all vectors in C are divisible by 4. A vector whose weight is divisible by 4 is also called d.e. An even code which is not doubly-even is called singly-even (s.e.). The author calls an even code balanced if it contains the same number of vectors whose weights are /spl equiv/0(mod 4) as those of weights /spl equiv/2(mod 4). Any s.e. self-orthogonal code is balanced. The author calls a coset balanced if either all weights in it are even and half are /spl equiv/0(mod 4), half /spl equiv/2(mod 4) or all weights are odd and half are /spl equiv/1(mod 4), half /spl equiv/3(mod 4).<<ETX>>