On the subspace distance of the subspace codes

Document Type : Research Paper

Authors

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

Abstract

Let Pq(n) be the set of all subspaces in the vector space Fqn. There is a subspace distance dS(U,V) between any two subspaces U and V. A subspace code is also a subset of Pq(n). It is known that dS(U,V)≥dH(ν(πU),ν(πV)), where π∈Sn, ν(U) denotes the pivot vector of E(U) and E(U) is the reduced row echelon form of the generator matrix of U. In this paper, we show that if E(U) and E(V) have at most one non-zero entry in each rows and each columns then the equality holds. Moreover, we introduce the sets GU,V={π∈Sn∣dS(U,V)=dH(ν(πU),ν(πV))} for any U,V∈Pq(n) and examine them in the spaces P2(4), P2(5), P2(6) and P3(4). It is shown that the groups 1, Z2, Z2×Z2, S3, S4 and 1, Z2, Z2×Z2, S3, D8, S3×Z2, S4, S5 appears between these sets in P2(4) and P2(5), respectively. Moreover, the groups 1, Z2, Z2×Z2, S3, D8, Z2×Z2×Z2, S3×Z2, D8×Z2, S4, S3×S3, S4×Z2, (S3×S3):2, S5, S6 and 1, Z2, Z2×Z2, S3, D8, S4 appears between these sets in P2(6) and P3(4), respectively.
 

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References

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Algebraic Structures and Their Applications
Volume 12, Issue 1
February 2025
Pages 65-76

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