Best Known (21, 21+4, s)-Nets in Base 2
(21, 21+4, 4097)-Net over F2 — Constructive and digital
Digital (21, 25, 4097)-net over F2, using
(21, 21+4, 4109)-Net over F2 — Digital
Digital (21, 25, 4109)-net over F2, using
- net defined by OOA [i] based on linear OOA(225, 4109, F2, 4, 4) (dual of [(4109, 4), 16411, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(225, 4109, F2, 3, 4) (dual of [(4109, 3), 12302, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(225, 4109, F2, 4) (dual of [4109, 4084, 5]-code), using
- 1 times truncation [i] based on linear OA(226, 4110, F2, 5) (dual of [4110, 4084, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(225, 4096, F2, 5) (dual of [4096, 4071, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(213, 4096, F2, 3) (dual of [4096, 4083, 4]-code or 4096-cap in PG(12,2)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- 1 times truncation [i] based on linear OA(226, 4110, F2, 5) (dual of [4110, 4084, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(225, 4109, F2, 4) (dual of [4109, 4084, 5]-code), using
- appending kth column [i] based on linear OOA(225, 4109, F2, 3, 4) (dual of [(4109, 3), 12302, 5]-NRT-code), using
(21, 21+4, 8189)-Net in Base 2 — Upper bound on s
There is no (21, 25, 8190)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 33 558526 > 225 [i]