We consider the [ q+1, q-3,5]q3 generalized doubly-extended Reed-Solomon code of codimension 4 as the code associated with the twisted cubic in the projective space PG(3,q). Basing on the point-plane incidence matrix of PG(3,q), we obtain the number of weight 3 vectors in all the cosets of the considered code. This allows us to classify the cosets by their weight distributions and to obtain these distributions. The weight of a coset is the smallest Hamming weight of any vector in the coset. For the cosets of equal weight having distinct weight distributions, we prove that the difference between the w-th components, 3 <; w ≤ q+1, of the distributions is uniquely determined by the difference between the 3-rd components. This implies an interesting (and in some sense unexpected) symmetry of the obtained distributions.

We consider the weight spectrum of a class of quasi-perfect binary linear codes with code distance 4. For example, extended Hamming code and Panchenko code are the known members of this class. Also, it is known that in many cases Panchenko code has the minimal number of weight 4 codewords. We give exact recursive formulas for the weight spectrum of quasi-perfect codes and their dual codes. As an example of application of the weight spectrum we derive a lower estimate for the conditional probability of correction of erasure patterns of high weights (equal to or greater than code distance).

A concept of locally optimal (LO) linear covering codes is introduced in accordance with the concept of minimal saturating sets in projective spaces over finite fields. An LO code is nonshortening in the sense that one cannot remove any column from a parity-check matrix without increasing the code covering radius. Several q/sup m/-concatenating constructions of LO covering codes are described. Taking a starting LO code as a "seed", such constructions produce an infinite family of LO codes with the same covering radius. The infinite families of LO codes are designed using minimal saturating sets as starting codes. New upper bounds on the length function are given. New extremal and classification problems for linear covering codes are formulated and investigated, in particular, the spectrum of possible lengths of LO codes including the greatest possible length. The complete computer classification of the minimal saturating sets in small geometries and of the corresponding LO codes is obtained.

Infinite families of linear codes with covering radius R=2, 3 and codimension tR+1 are constructed on the base of starting codes with codimension 3 and 4. Parity-check matrices of the starting codes are treated as saturating sets in projective geometry that are obtained by computer search using projective properties of objects. Upper bounds on the length function and on the smallest sizes of saturating sets are given.

Let [n,n-r]/sub q/R denote a linear code over F/sub q/ with length n, codimension r, and covering radius R. We use a modification of constructions of [2q+1, 2q-3]/sub q/2 and [3q+1, 3q-5]/sub q/3 codes (q/spl ges/5) to produce infinite families of good codes with covering radius 2 and 3 and codimension tR.

New constructions of linear nonbinary codes with covering radius R=2 are proposed. They are in part modifications of earlier constructions by the author and in part are new. Using a starting code with R=2 as a "seed" these constructions yield an infinite family of codes with the same covering radius. New infinite families of codes with R=2 are obtained for all alphabets of size q/spl ges/4 and all codimensions r/spl ges/3 with the help of the constructions described. The parameters obtained are better than those of known codes. New estimates for some partition parameters in earlier known constructions are used to design new code families. Complete caps and other saturated sets of points in projective geometry are applied as starting codes, A table of new upper bounds on the length function for q=4, 5.7, R=2, and r/spl les/24 is included.

Constructions of nonlinear covering codes are given. Using any nonlinear starting code of covering radius R/spl ges/2 these constructions form an infinite family of codes with the same covering radius. A nonlinear code is treated as a union of cosets of a linear code. New infinite families of nonlinear covering codes are obtained. Concepts of R,l-objects, R,l-partitions, and R,l-length are described for nonlinear codes.

In Davydov (1990), constructions of linear binary covering codes were considered. In the present paper, constructions and techniques of the earlier paper are developed and modified for q-ary linear nonbinary covering codes, q/spl ges/3, and new constructions are proposed. The described constructions design an infinite family of codes with covering radius R based on a starting code of the same covering radius. For arbitrary R/spl ges/2, q/spl ges/3, new infinite families of nonbinary covering codes with "good" parameters are obtained with the help of an iterative process when constructed codes are the starting codes for the following steps. The table of upper bounds on the length function for codes with q=3, R=2, 3, and codimension up to 24 is given. The author proposes to use saturated sets of points in projective geometries over finite fields as parity check matrices of starting codes. New saturated sets are obtained.

Presents constructions and infinite families of binary linear covering codes with covering radii R=2,3,4. Using these codes, the authors obtain a table of constructive upper bounds on the length function l(r,R) for r/spl les/64 and R=2,3,4, where l(r, R) is the smallest length of a binary linear code with given codimension r and covering radius R. They obtain also upper bounds on l(r, R) for r=21, 28, R=5. Parameters of the constructed codes are better than parameters of previously known codes.>

The Pi code constructed by V.I. Panchenko (1987) is studied. The Pi code as an alternative to the Hamming code in the class of single-error-correcting and double-error-detecting codes (SEC-DED codes) is also considered. The Pi code has a smaller number of words of weight 4 and provides a larger probability of triple-independent-error detection than the shortened Hamming code with the same parameters. Shortening algorithms for the Pi code are proposed, and parity check matrices of the (39,32), (72,64), (137,128) shortened Pi codes are constructed. The obtained codes can detect byte errors of length 4. The parity check matrices of the Pi code have more 1's in rows than corresponding matrices of the Hamming code. The Pi code is a reasonable alternative to the Hamming code in the class of SEC-DED codes.>

Infinite families of linear binary codes with covering radius R=2 and minimum distance d=3 and d=4 are given. Using the constructed codes with d=3, R=2, families of covering codes with R>2 are obtained. The parameters of many constructed codes with R>