Best Known (7−4, 7, s)-Nets in Base 256
(7−4, 7, 32769)-Net over F256 — Constructive and digital
Digital (3, 7, 32769)-net over F256, using
- net defined by OOA [i] based on linear OOA(2567, 32769, F256, 4, 4) (dual of [(32769, 4), 131069, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2567, 65536, F256, 4) (dual of [65536, 65529, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(2565, 65536, F256, 3) (dual of [65536, 65531, 4]-code or 65536-cap in PG(4,256)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
(7−4, 7, 65538)-Net over F256 — Digital
Digital (3, 7, 65538)-net over F256, using
- net defined by OOA [i] based on linear OOA(2567, 65538, F256, 4, 4) (dual of [(65538, 4), 262145, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(2567, 65538, F256, 3, 4) (dual of [(65538, 3), 196607, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2567, 65536, F256, 4) (dual of [65536, 65529, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(2565, 65536, F256, 3) (dual of [65536, 65531, 4]-code or 65536-cap in PG(4,256)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
- appending kth column [i] based on linear OOA(2567, 65538, F256, 3, 4) (dual of [(65538, 3), 196607, 5]-NRT-code), using
(7−4, 7, 1488725)-Net in Base 256 — Upper bound on s
There is no (3, 7, 1488726)-net in base 256, because
- the generalized Rao bound for nets shows that 256m ≥ 72057 668825 212786 > 2567 [i]