Fix integers $e\ge2$ and $r\ge1$. In this paper we study the $r$-th generalized covering radius $\rho_r\left(BCH(e,m)\right)$ of the binary primitive $e$-error-correcting BCH code $BCH(e,m)$. By using an algebraic-geometric reformulation of the covering problem together with an explicit Lang-Weil estimate, we prove tha...
We develop a multi-view linear-programming framework for locally recoverable codes with arbitrary fixed availability $a$. For any retained order $1 \le s \le a$, the selected helper sets, the recovered coordinate, and their complement form an $(s+2)$-part partition. Recording the Hamming distance on all blocks preserve...
Ming-Hsuan Kang, Mao-Sheng Xiong, Yun-Hsuan Hsieh et al.· 0 citations
We derive linear programming (LP) bounds on the minimum distance of binary and ternary linear complementary dual (LCD) codes by imposing arithmetic constraints on their weight enumerators. Special values of the weight enumerator give finitely many Gauss phases, each of which yields linear equations in the ordinary weig...
A polynomial-size linear programming bound for locally recoverable codes with locality parameters $(r,\delta)$, without assuming linearity is given, which has polynomially many variables and its optimum dominates the ordinary Delsarte bound.
For $q\in\set{2,3}$, we show that a $k$-dimensional linear code over the finite field $\F_q$ of order $q$ is linear complementary dual (LCD) exactly when one root-of-unity value of its weight enumerator has magnitude $q^{k/2}$. We convert the phase of this value, together with the parity type in the binary case, into e...
New Delsarte-type linear programming bounds for $q$-ary $(r,\delta)$-locally recoverable codes (LRCs) are derived with three attributes: the variable set is comparable in size to that of the classical Delsarte LP, and the convex-hull LP yields improvements not captured by the previous LP and often sharpens the shorteni...
