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Circular-Shift-Based Vector Linear Network Coding and Its Application to Array Codes
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DOI：10.1109/TIT.2025.3622163.png)
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Abstract
En 中文
linear network coding (LNC) is a class of vector LNC with local encoding kernels selected from cyclic permutation matrices, so that it has low coding complexities. However, it is insufficient to exactly achieve the capacity of a multicast network, so the data units transmitted along the network need to contain redundant symbols, which affects the transmission efficiency. In this paper, as a variation of circular-shift LNC, we introduce a new class of vector LNC over arbitrary GF(p), called circular-shift-based vector LNC, which is shown to be able to exactly achieve the capacity of a multicast network. The set of local encoding kernels in circular-shift-based vector LNC is nontrivially designed such that it is closed under multiplication by elements in itself. It turns out that the coding complexity of circular-shift-based vector LNC is comparable to and, in some cases, even lower than that of circular-shift LNC. The new results in the formulation of circular-shift-based vector LNC further facilitates us to characterize and design Vandermonde circulant maximum distance separable (MDS) array codes, which are built upon the structure of Vandermonde matrices and circular-shift operations. We prove that for r >= 2, the largest possible k for an L-dimensional (k + r, k) Vandermonde circulant p-ary MDS array code is p(mL)-1, where L is an integer co-prime with p, and m(L) represents the multiplicative order of p modulo L. For r is an element of {2, 3}, we introduce two new types of (k + r, k) p-ary array codes that can achieve the largest k = p(mL)-1. For the special case that p = 2, we propose scheduling encoding algorithms for the 2 new codes, so that the encoding complexity not only asymptotically approaches the optimal 2 XORs per original data bit, but also slightly outperforms the encoding complexity of other known Vandermonde circulant MDS array codes with largest k = 2(mL)-1.
Keywords:
Vectors
Codes
Encoding
Kernel
Complexity theory
Arrays
Receivers
Network coding
Linear codes
Decoding
circular-shift linear code
vector linear code
array codes
Vandermonde
encoding complexity
Journal
I
IF:
2.9
Papers:
317
Citations:
0