Geodesic
Coordinate transitivity of a class of extended perfect codes and their SQS
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1451-1462

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We continue the study of the class of binary extended perfect propelinear codes constructed in the previous paper and consider their permutation automorphism (symmetry) groups and Steiner quadruple systems. We show that the automorphism group of the SQS of any such code coincides with the permutation automorphism group of the code. In particular, the isomorphism classes of these SQS's are complete invariants for the isomorphism classes of these codes. We obtain a criterion for the point transitivity of the automorphism group of SQS of proposed codes in terms of $\mathrm{GL}$-equivalence (similar to EA-type equivalence for permutations of $F^r$). Based on these results we suggest a new construction for coordinate transitive and neighbor transitive extended perfect codes.
Keywords: extended perfect code, concatenation construction, neighbor transitive code, transitive action, regular subgroup, isomorphism problem, transitive Steiner quadruple system, coordinate transitive code.
Mots-clés : transitive code 
@article{SEMR_2020_17_a74,
     author = {I. Yu. Mogilnykh and F. I. Solov'eva},
     title = {Coordinate transitivity of a class of extended perfect codes and their {SQS}},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1451--1462},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a74/}
}
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I. Yu. Mogilnykh; F. I. Solov'eva. Coordinate transitivity of a class of extended perfect codes and their SQS. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1451-1462. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a74/