New extremal binary [44,22,8] codes

One of the problems in coding theory is constructing self-dual codes whose weight enumerators are not yet known to exist. We use the concept of neighbors and construct extremal binary self-dual [44,22,8] codes whose weight enumerators are not yet known to exist.     

The true dimension of certain binary Goppa codes

By applying a result from algebraic geometry due to E. Bombieri (Amer. J. Math., vol.88, p.71-105, 1966), the true dimension of certain binary Goppa codes is calculated. The results lead in many cases to an improvement of the usual lower bound for the dimension     

Weight distribution of a class of binary block codes formed from RCPC codes

In this paper, we study the weight enumerator and the numerical performance of a class of binary linear block codes formed from a family of rate-compatible punctured convolutional (RCPC) codes. Also, we present useful numerical results for a well-known family of RCPC codes.     

New bounds on binary linear codes of dimension eight (Corresp.)

Letn(k,d)be the smallest integernsuch that a binary linear code of lengthn, dimensionk, and minimum distance at leastdexists. New results are given that improve the best previously known bounds onn(8,d).     

Weight distributions of some classes of binary cyclic codes (Corresp.)

Leth_1(x)h_2(x)be the parity-check polynomial of a binary cyclic code. This correspondence presents a formula for decomposing words in the code as sums of multiples of words in the codes whose parity-check polynomials areh_1(x)andh_2(x). This decomposition provides information about the weight distribution of the code.     

Existence of some extremal self-dual codes

Two new singly-even extremal, self-dual codes are constructed: a [52,26,10] code and a [54,27,10] code     

Circulant based extremal additive self-dual codes over GF(4)

It is well known that the problem of finding stabilizer quantum-error-correcting codes (QECC) is transformed into the problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product. Our purpose is to classify the extremal additive circulant self-dual codes of lengths up to 15, and construct good codes for lengths 16/spl les/n/spl les/27. We also classify the extremal additive 4-circulant self-dual codes of lengths 4,6,8,12,14, and 16 and most codes of length 10, and construct good codes of even lengths up to 22. Furthermore, we classify the extremal additive bordered 4-circulant self-dual codes of lengths 3,5,7,9,11,13,15, and 17, and construct good codes for lengths 19,21,23, and 25. We give the current status of known extremal (or optimal) additive self-dual codes of lengths 12 to 27.     

The weight hierarchies of some product codes

We determine the weight hierarchies of the product of an n-tuple space and an arbitrary code, the product of an m-dimensional even-weight code and the [24,12,8] extended Golay code, and the product of an m-dimensional even-weight code and the [8,4,4] extended Hamming code. The conjecture d_r=d*_r is proven for all three cases     

Optimum codes of dimension 3 and 4 over GF(4)

n_q(k,d), the length of a q-ary optimum code for given k and d, for q=4 and k=3, 4 is discussed. The problem is completely solved for k=3, and the exact value of n₄(4,d) is determined for all but 52 values of d     

New binary codes

In this paper constructions are given for combining two, three, or four codes to obtain new codes. The Andryanov-Saskovets construction is generalized. It is shown that the Preparata double-error-correcting codes may be extended by about (block length)^{1/2}symbols, of which only one is a check symbol, and thate-error-correcting BCH codes may sometimes be extended by (block !ength)^{1/e}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 withd/n = frac{1}{3}, the first three being the(24,2^12, 8)Golay code, a(48,2^15, 16)code, and a(96,2^18, 32)code. Most of the codes given have more codewords than any comparable code previously known to us.     

New binary codes

We report on the construction of many binary linear codes improving the tables of the best known codes. We obtained our results by applying the following known constructions: shortening and puncturing codes by analyzing their duals; and transferring a [64,8,43] code over GF(4) into a binary code and applying various constructions to the resulting code     

Soft decoding performance of extremal self-dual codes

Performance of soft decoded extremal self-dual codes of lengths 8 to 72 are obtained over the Gaussian channel. The results indicate that for decoded bit-error rates below 10^-3, which is the main region of interest for coding application, substantial coding gains can be obtained by using extremal self-dual codes.     

New extremal self-dual codes of length 62 and related extremalself-dual codes

New extremal self-dual codes of length 62 are constructed with weight enumerators of three different types. Two of these types were not represented by any known code up till now. All these codes possess an automorphism of order 15. Some of them are used to construct extremal self-dual codes of length 60 by the method of subtracting. By additional subtracting, an extremal self-dual [58, 29, 10] code was obtained having a weight enumerator which does not correspond to any code known so far     

Self-synchronizing codes derived from binary cyclic codes

The sensitivity of a binary block code to loss of synchronism (misplacement of the "commas" separating codewords) can be characterized by a pair of numbers[s, delta]such that any synchronization slip of s bits or less produces an overlap sequence differing from a legitimate codeword in at leastdeltaplaces. This definition is broader than that of comma freedom of indexdelta, which is included as the special case of s equal to the integer part of half the code block length. For codes having the slip-detecting characteristic[s, delta]there exists the possibility of implementation to restore synchronism during an interval relatively free from bit errors. It is shown that certain error-correcting binary cyclic block codes can be altered to obtain the characteristic[s, delta]by the addition of a fixed binary vector to each codeword prior to transmission. These altered cyclic codes retain the full error-correcting power of the original cyclic codes. An error-detecting/correcting data format providing protection against the acceptance of misframed data is thus obtained without the insertion of special synchronizing sequences into the bit stream.     

Orthogonality of binary codes derived from Reed-Solomon codes

The author provides a simple method for determining the orthogonality of binary codes derived from Reed-Solomon codes and other cyclic codes of length 2^m-1 over GF(2^m) for m bits. Depending on the spectra of the codes, it is sufficient to test a small number of single-frequency pairs for orthogonality, and a pair of bases may be tested in each case simply by summing the appropriate powers of elements of the dual bases. This simple test can be used to find self-orthogonal codes. For even values of m, the author presents a technique that can be used to choose a basis that produces a self-orthogonal, doubly-even code in certain cases, particularly when m is highly composite. If m is a power of 2, this technique can be used to find self-dual bases for GF(2 ^m). Although the primary emphasis is on testing for self orthogonality, the fundamental theorems presented apply also to the orthogonality of two different codes     

Optimal binary linear codes and Z4-linearity

We give necessary and sufficient conditions for a binary linear code to be Z₄-linear. Especially we treat optimal, binary linear codes and determine all such codes with minimum weight less or equal to six which are Z₄-linear     

Nonrandom binary superimposed codes

A binary superimposed code consists of a set of code words whose digit-by-digit Boolean sums(1 + 1 = 1)enjoy a prescribed level of distinguishability. These codes find their main application in the representation of document attributes within an information retrieval system, but might also be used as a basis for channel assignments to relieve congestion in crowded communications bands. In this paper some basic properties of nonrandom codes of this family are presented, and formulas and bounds relating the principal code parameters are derived. Finally, there are described several such code families based upon (1)q-nary conventional error-correcting codes, (2) combinatorial arrangements, such as block designs and Latin squares, (3) a graphical construction, and (4) the parity-check matrices of standard binary error-correcting codes.     

Permutation-decodable binary cyclic codes

The letter develops bounds on the information rates of certain binary cyclic codes so that they are s-step permutation decodable (s=2, 3, 4).     

Cubic self-dual binary codes

We study binary self-dual codes with a fixed point free automorphism of order three. All binary codes of that type can be obtained by a cubic construction that generalizes Turyn's. We regard such "cubic" codes of length 3/spl lscr/ as codes of length /spl lscr/ over the ring F/sub 2//spl times/F/sub 4/. Classical notions of Type II codes, shadow codes, and weight enumerators are adapted to that ring. Two infinite families of cubic codes are introduced. New extremal binary codes in lengths /spl les/ 66 are constructed by a randomized algorithm. Necessary conditions for the existence of a cubic [72,36,16] Type II code are derived.     

Automorphisms of codes with applications to extremal doubly even codes of length 48

General results on automorphisms of self-dual binary codes are given. These results are applied to the study of extremal self-dual doubly even binary codes of length48. The main theorem proved is that an extremal self-dual doubly even code of length48with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code. Interesting constructions of the binary extended Golay code as well as a conjecture about a possible connection between an extremal self-dual doubly even code of length72and an extremal quaternary code of length24arc yielded by techniques used in the proof.