In this paper, we focus on Singleton-optimal$(r,\delta)$LRCs with disjoint local repair groups. We provide an improved bound for the length of q-ary Singleton-optimal$(r,\delta)$LRCs based on the parity-check matrix approach. Specifically, for$d \geq 3\delta $, we prove that$n\le O(q^{\delta })$when$d-3\delta \lt r\le d-2\delta +1$. We also show that the code length$n\le q+\delta +2$when$r=2$and$d=3\delta +2$. We present a sufficient and necessary condition for the existence of Singleton-optimal$(n,k,d;r,\delta)$LRCs with disjoint local repair groups, where the minimum distance satisfies$3\delta +1\le d \le 3\delta +2$and locality$r=2$. This condition imposes an upper bound on the code length,$n\le O(q^{2})$, and indicates the existence of a code length approximately given by$n\approx \sqrt {2}q$when$d=3\delta +1$and$r=2$. Finally, we utilize blocking sets to provide a general construction of Singleton-optimal$(n,k,d=2\delta +2,r=2,\delta)$LRC with code length$n\approx O\left ({{q^{\frac {h+1}{h}}}}\right)$for any$h\ge 3$. To the best of our knowledge, this is the first family of Singleton-optimal$(n,k,d=2\delta +2,r=2,\delta)$LRC with super-linear code length.

An$(n, k, d; r)_{q}$-locally repairable code (LRC) is called a Singleton-optimal LRC if it achieves the Singleton-type bound. Analogous to the classical MDS conjecture, the maximal length problem of Singleton-optimal LRCs has attracted a lot of attention in recent years. In this paper, we give an improved upper bound for the length of q-ary Singleton-optimal LRCs with disjoint repair groups such that$(r+1)\mid n$based on the parity-check matrix approach. In particular, for any Singleton-optimal$(n, k, d; r)_{q}$-LRCs, we show that: 1)$n\le q+d-4$, when$r=2$and$d=3e+8$with$e\ge 0$; 2)$n\leq (r+1)\left \lfloor {{\frac {2(q^{2}+q+1)}{r(r+1)} +e+1}}\right \rfloor $, when$d\ge 8$and$\max \left \{{{3,\frac {d-e-6}{e+1}}}\right \}\le r\le \frac {d-e-3}{e+1}$for any$0\le e\le \left \lfloor {{\frac {d-6}{4} }}\right \rfloor $. Furthermore, we establish equivalent connections between the existence of Singleton-optimal$(n,k,d;r)_{q}$-LRCs for$d=6, r=3$and$d=7, r=2$with disjoint repair groups and some subsets of lines in finite projective space with certain properties. Consequently, we prove that the length of q-ary Singleton-optimal LRCs with minimum distance$d=6$and locality$r=3$is upper bounded by$O(q^{1.5})$. We construct Singleton-optimal$(8\le n\le q+1,k,d=6,r=3)_{q}$-LRC with disjoint repair groups such that$4\mid n$and determine the exact value of the maximum code length for some specific q. We also prove the existence of$(n, k, d=7; r=2)_{q}$-Singleton-optimal LRCs for$n \approx \sqrt {2}q$.

The famous Barnes–Wall lattices can be obtained by applying Construction D to a chain of Reed–Muller codes. By applying Construction${\text {D}}^{\text {(cyc)}}$to a chain of extended cyclic codes sandwiched between Reed–Muller codes, Hu and Nebe (J. London Math. Soc.(2)101 (2020) 1068-1089) constructed new series of universally strongly perfect lattices sandwiched between Barnes–Wall lattices. In this paper, we first extend their construction to generalized Reed–Muller codes, and then explicitly determine the minimum vectors of those new sandwiched Reed–Muller codes for some special cases.

In this paper, we study a class of linear codes defined by characteristic functions of certain subsets of a finite field. We derive a sufficient and necessary condition for such a code to be a minimal linear code by a character-theoretical approach. We obtain new three-weight or four-weight minimal linear codes that do not satisfy the Ashikhmin-Barg condition by using partial difference sets. We show that our construction yields minimal linear codes that do not arise from cutting vectorial blocking sets, and also discuss their applications in secret sharing schemes.