Structural properties of binary propelinear codes

A concatenation construction for propelinear perfect codes from regular subgroups of $\mathrm {GA}(r,2)$

“…From Lemma 2 we see that the dimension of the kernel of a code S H,τ of length 2 r+1 − r − 2 is at least 2 r+1 − 2r − 2. If the dimension of the kernel of S H,τ is 2 r+1 − 2r − 2 then the code S H,τ could not be obtained by the Mollard construction for propelinear codes with large ranks, see [2]. Recall that two codes of length n are called equivalent if there is an automorphism of F n that maps one code to another.…”

“…The values for (l, k), where l is the distension and k is the deficiency of the permutations induced by the automorphisms of the regular subgroups of GA(r, 2) r = 3 (0,0),(1,2),(2,3),(3,3) r = 4 (0,0),(1,2),(2,3), (2,4), (3,3), (3,4),(4,3) r = 5 (0,0), (1,2), (1,4), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5),(4,3),(4,4),(4,5), (5,4),(5,5) Table 1.…”

“…Classical propelinear perfect codes are Z 4 -linear [9] and Z 2 Z 4 -linear codes [5]. It is known that propelinear perfect codes can be obtained by the Plotkin and the Vasil'ev constructions [2]. In [3] all transitive codes from [21] found by a representation via the Phelps construction were proved to be propelinear.…”

“…The codes from [21] were later generalized by Krotov and Potapov in [11] who utilized quadratic functions in the Vasil'ev construction. Note that the Vasil'ev construction was generalized by the Mollard construction for propelinear perfect codes [2]. The approaches of [21,3,11] gave exponential classes of propelinear codes (the best lower bound was obtained in [11]) but all these codes are of small rank n − log(n + 1) + 1, where n is the length of the codes.…”

On separability of the classes of homogeneous and transitive perfect binary codes

“…Both questions have been answered positively. The answer to the first question was obtained in [3], where it was proved that the well-known binary Best's code of length 10 with code distance 4, being transitive, is not propelinear. The existence of an infinite series of transitive nonpropelinear perfect codes was proved in [4]: It should be noted that in [4] it was proved that only one of the 201 equivalence classes of transitive perfect codes of length 15 is not propelinear.…”