Circular-Shift-Based Vector Linear Network Coding and Its Application to Array Codes
Circular-shift 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 t...
Gespeichert in:
| Veröffentlicht in: | IEEE transactions on information theory Jg. 71; H. 12; S. 9413 - 9431 |
|---|---|
| Hauptverfasser: | , , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
IEEE
01.12.2025
|
| Schlagworte: | |
| ISSN: | 0018-9448, 1557-9654 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Zusammenfassung: | Circular-shift 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(<inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>), 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 <inline-formula> <tex-math notation="LaTeX">r \geq 2 </tex-math></inline-formula>, the largest possible <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> for an <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula>-dimensional <inline-formula> <tex-math notation="LaTeX">(k+r, k) </tex-math></inline-formula> Vandermonde circulant <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>-ary MDS array code is <inline-formula> <tex-math notation="LaTeX">p^{m_{L}}-1 </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula> is an integer co-prime with <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">m_{L} </tex-math></inline-formula> represents the multiplicative order of <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula> modulo <inline-formula> <tex-math notation="LaTeX">L </tex-math></inline-formula>. For <inline-formula> <tex-math notation="LaTeX">r \in \{2, 3\} </tex-math></inline-formula>, we introduce two new types of <inline-formula> <tex-math notation="LaTeX">(k+r, k)~p </tex-math></inline-formula>-ary array codes that can achieve the largest <inline-formula> <tex-math notation="LaTeX">k = p^{m_{L}}-1 </tex-math></inline-formula>. For the special case that <inline-formula> <tex-math notation="LaTeX">p = 2 </tex-math></inline-formula>, 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 <inline-formula> <tex-math notation="LaTeX">k = 2^{m_{L}} - 1 </tex-math></inline-formula>. |
|---|---|
| ISSN: | 0018-9448 1557-9654 |
| DOI: | 10.1109/TIT.2025.3622163 |