Circular-shift-based Vector Linear Network Coding and Its Application to Array Codes
Sheng Jin, Zhe Zhai, Qifu Tyler Sun, Zongpeng Li
TL;DR
This work addresses achieving the multicast capacity for $L$-dimensional vector LNC over GF($p$) while keeping coding complexity low. It introduces circular-shift-based vector LNC with local kernels from ${\cal C}=\{\boldsymbol{P}(\sum_{j=0}^{L-1} a_j \mathbf{C}_L^j)\boldsymbol{Q}: a_j\in\mathrm{GF}(p)\}$, and designs $\boldsymbol{P},\boldsymbol{Q}$ so that ${\cal C}$ is closed under multiplication and consists of invertible matrices, enabling exact capacity achievement. The framework establishes a direct link to Vandermonde circulant MDS array codes, showing that for $r\ge 2$ the largest $k$ in an $L$-dimensional $(k+r,k)$ p-ary MDS array code is $p^{m_L}-1$ (with $m_L$ the multiplicative order of $p$ modulo $L$ and $\beta$ a primitive $L$-th root of unity), and provides two new $(k+r,k)$ codes for $r\in\{2,3\}$ attaining $k=p^{m_L}-1$. For the binary case ($p=2$) and prime $L$, scheduling algorithms further reduce encoding complexity toward the optimum $2$ XORs per data bit, illustrating practical gains for storage-oriented array codes. Overall, the work unifies vector LNC and array-code design, enabling capacity-achieving, low-complexity schemes and enabling larger, efficient Vandermonde circulant MDS array codes.
Abstract
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($p$), 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, lower than that of circular-shift LNC. The new results in 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 $r \geq 2$, the largest possible $k$ for an $L$-dimensional $(k+r, k)$ Vandermonde circulant $p$-ary MDS array code is $p^{m_L}-1$, where $L$ is an integer co-prime with $p$, and $m_L$ represents the multiplicative order of $p$ modulo $L$. For $r = 2, 3$, we introduce two new types of $(k+r, k)$ $p$-ary array codes that achieves the largest $k = p^{m_L}-1$. 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 $k = p^{m_L}-1$.