Temel Kavramlar
This paper constructs binary linear sum-rank-metric codes with matrix size 2×2 from quaternary BCH and Goppa codes, and presents fast decoding algorithms for these codes.
Özet
The paper focuses on the construction and decoding of binary linear sum-rank-metric codes with matrix size 2×2.
Key highlights:
Constructs binary linear sum-rank-metric codes with matrix size 2×2 from quaternary BCH and Goppa codes. These codes are called BCH-type and Goppa-type sum-rank-metric codes.
Provides a reduction of the decoding in the binary sum-rank-metric space to the decoding in the Hamming-metric space. This reduction is applied to the BCH and Goppa-type codes.
Presents fast decoding algorithms for the BCH and Goppa-type binary linear sum-rank-metric codes, with complexity O(ℓ^2) operations in the field F4.
Constructs asymptotically good sequences of binary linear sum-rank-metric codes with matrix size 2×2 from Goppa codes, which can be decoded efficiently.
Compares the constructed codes with the existing sum-rank BCH codes, showing that the new codes have larger dimensions for the same minimum sum-rank distances.
İstatistikler
wtsr(a2x + a1x^2) = 2wtH(a1) + 2wtH(a2) - 3|I|, where I = supp(a1) ∩ supp(a2).
The minimum sum-rank distance of SR(C1, C2) is at least max{min{d1, 2d2}, min{d2, 2d1}}.
Alıntılar
"Sum-rank-metric codes have wide applications in multishot network coding, see [35, 43, 47], space-time coding, see [50], and coding for distributed storage, see [14, 33, 36]."
"Fast algebraic decoding and list-decoding algorithms for BCH codes, Goppa codes and algebraic geometry codes in the Hamming-metric are well-developed, see [27, Chapter 5], [16,19] and [24]."