17 does not divide the order of the group of a (72,36,16) doubly even code (Corresp.)

It is an interesting open question whether an extremal (72, 36, 16) doubly even code C exists. In [3] the odd prime numbers which can divide the order of the group of C were determined. The largest of these, 23, was eliminated by finding weight 12 vectors in 384 codes [8]. The next largest prime remaining is 17. It is shown that 17 is also not possible by reducing the problem to the consideration of16.17^{3}codes and then finding a weight 12 vector, by computer, in each of these codes.     

A [72,36,16] doubly even code does not have an automorphism of order 11 (Corresp.)

We prove that there does not exist a[72,36,16]doubly even code with an automorphism of order11. If such a code exists, it can be decomposed as a direct sum of two codes; one can be viewed as a self-dual[12,6,4]binary code and the other a self-dual[6,3,4]code overF_{1024}. This is shown to be impossible.     

A binary extremal doubly even self-dual code (64,32,12) obtained from an extended Reed-Solomon code over Fl6 (Corresp.)

The binary image of an extended Reed-Solomon code overF_{16}relative to a trace-orthogonal basis ofF_{l6}is an extremal doubly even self-dual code(64,32,12).     

On the automorphism group of a doubly-even (72,36,16) code

We prove that a putative doubly-even (72,36,16) code has no automorphism of order 3 with fixed points. Moreover, such a code has no automorphism of an odd order greater than 7. Eventually, we prove that 25 and 49 do not divide the order of its automorphism group.     

A self-dual even (96, 48, 16) code (Corresp.)

IfC_1, C_2are two codes of the same length, letC_1 neq C_2 = {(alpha + eta,beta + eta, alpha + beta + eta) mid alpha, beta in C_1, eta in C_2}. TakingC_1andC_2to be two extended quadratic residue codes of length 32, it is shown thatC_1 neq C_2is a (96,48,16) self-dual even code with all weights divisible by 4.     

On a (21,11,6) binary code (Corresp.)

It is shown that the (21,11,6) binary linear code obtained from BCH codes and a single parity check code by constructionXis equivalent to a difference-set cyclic code.     

Binary even [25,15,6] codes do not exist (Corresp.)

That binary even[25,15,6]codes do not exist is proved by passing from coding theory to{bf F}_{2}-affine geometry; all25-sets in{bf F}^{9}_{2}are shown to contain affine planes.     

A characterization of a (56,28) extremal self-dual code (Corresp.)

A(54,27)code is characterized in terms of a27 times 27matrix made up of circulant3 times 3matrices. The minimum distance of this code is found to be11, and its weight distribution is computed. The(54,27)d=11code is extended to a(56,28)d=12external self-dual code. A(54,27)d=10self-dual code is also described in terms of the incidence matrix of a(27,13,6)group difference set defined on GF(3^{3}).     

Algebraic decoding of the (23,12,7) Golay code (Corresp.)

It is shown that the algebraic method for decoding three-error-correcting BCH codes is also applicable to complete decoding of the(23,12,7)Golay code.     

A case where interference does not reduce capacity (Corresp.)

It is shown that, under certain conditions, two strongly interfering communication links with additive white Gaussian noise can achieve rates as high as would be achievable without this interference.     

Gray code weighting system (Corresp.)

Contrary to what has been believed, it is found that weights can be associated with different bit positions of the standard binary reflected Gray code. Therefore, two algorithms are devised for direct Gray-to-decimal and decimal-to-Gray conversions. These algorithms are much simpler than old algorithms which involve conversions to and from binary code and exclusive-OR operations.     

The separability of Gray code (Corresp.)

It is proved that, when binary strings are Gray coded into multilevel signals, to cause an error ofmbits, the multilevel signal must have an error of more than2^m /3. This generalizes the well-known result that a one-level error in the signal generates only a one-bit error in the binary string.     

The second order Reed-Muller code of length 64 has covering radius 18 (Corresp.)

The assertion of the title is proved by first showing that the covering radius ofR(2, 6)cannot exceed18and then constructing a weight18coset leader.     

The weight-distribution of a coset of a linear code (Corresp.)

The coseta + Aof the linear codeAhas weight distribution related to that ofA^{perp}and ofA^{perp} cap a^{perp}.     

Feedback does not affect the reliability function of a DMC at rates above capacity (Corresp.)

Asymptotically coincident upper and lower bounds on the exponent of the largest possible probability of correct decoding for block codes of any given rate above capacity have been given by Dueck and Körner. Their method is extended to show that the same bounds also hold in the presence of noiseless feedback.