Propelinear 1-Perfect Codes From Quadratic Functions

Transitive 1-perfect codes from quadratic functions

Ranks of propelinear perfect binary codes

It is proven that for any numbers n = 2 m − 1, m ≥ 4 and r, such that n − log(n + 1) ≤ r ≤ n excluding n = r = 63, n = 127, r ∈ {126, 127} and n = r = 2047 there exists a propelinear perfect binary code of length n and rank r.

Binary quasi-perfect linear codes from APN quadratic functions

A mapping f from F2m to itself is almost perfect nonlinear (APN) if its directional derivatives in nonzero directions are all 2-to-1. Let Cf be the binary linear code of length 2 − 1, whose parity check matrix has its j-th column [ π f(π) ] , where π is a primitive element in F2m and j = 0, 1, · · · , 2 − 2. For m ≥ 3 and any quadratic APN function f(x) = ∑m−1 i,j=0 ai,jx 2+2 , ai,j ∈ F2m , it ...

On the Number of Nonequivalent Propelinear Extended Perfect Codes

The paper proves that there exists an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov (2007) are propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We investigate the ...

Translation-invariant propelinear codes

A class of binary group codes is investigated. These codes are the propelinear codes, deened over the Hamming metric space F n , F = f0; 1g, with a group structure. Generally, they are neither abelian nor translation invariant codes but they have good algebraic and com-binatorial properties. Linear codes and Z 4-linear codes can be seen as a subclass of prope-linear codes. Exactly, it is shown ...