Best Known (45−12, 45, s)-Nets in Base 7
(45−12, 45, 403)-Net over F7 — Constructive and digital
Digital (33, 45, 403)-net over F7, using
- net defined by OOA [i] based on linear OOA(745, 403, F7, 12, 12) (dual of [(403, 12), 4791, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(745, 2418, F7, 12) (dual of [2418, 2373, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(745, 2421, F7, 12) (dual of [2421, 2376, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(741, 2401, F7, 12) (dual of [2401, 2360, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 20, F7, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,7)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(745, 2421, F7, 12) (dual of [2421, 2376, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(745, 2418, F7, 12) (dual of [2418, 2373, 13]-code), using
(45−12, 45, 2591)-Net over F7 — Digital
Digital (33, 45, 2591)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(745, 2591, F7, 12) (dual of [2591, 2546, 13]-code), using
- 182 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 37 times 0, 1, 137 times 0) [i] based on linear OA(741, 2405, F7, 12) (dual of [2405, 2364, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(741, 2401, F7, 12) (dual of [2401, 2360, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(737, 2401, F7, 11) (dual of [2401, 2364, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 182 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 37 times 0, 1, 137 times 0) [i] based on linear OA(741, 2405, F7, 12) (dual of [2405, 2364, 13]-code), using
(45−12, 45, 1087188)-Net in Base 7 — Upper bound on s
There is no (33, 45, 1087189)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 107 007370 534929 287704 987361 055861 000505 > 745 [i]