Best Known (68, 91, s)-Nets in Base 7
(68, 91, 688)-Net over F7 — Constructive and digital
Digital (68, 91, 688)-net over F7, using
- 3 times m-reduction [i] based on digital (68, 94, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 47, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- trace code for nets [i] based on digital (21, 47, 344)-net over F49, using
(68, 91, 4912)-Net over F7 — Digital
Digital (68, 91, 4912)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(791, 4912, F7, 23) (dual of [4912, 4821, 24]-code), using
- 105 step Varšamov–Edel lengthening with (ri) = (1, 104 times 0) [i] based on linear OA(790, 4806, F7, 23) (dual of [4806, 4716, 24]-code), using
- trace code [i] based on linear OA(4945, 2403, F49, 23) (dual of [2403, 2358, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4945, 2401, F49, 23) (dual of [2401, 2356, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4943, 2401, F49, 22) (dual of [2401, 2358, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(4945, 2403, F49, 23) (dual of [2403, 2358, 24]-code), using
- 105 step Varšamov–Edel lengthening with (ri) = (1, 104 times 0) [i] based on linear OA(790, 4806, F7, 23) (dual of [4806, 4716, 24]-code), using
(68, 91, 6718943)-Net in Base 7 — Upper bound on s
There is no (68, 91, 6718944)-net in base 7, because
- 1 times m-reduction [i] would yield (68, 90, 6718944)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 11450 491542 779306 144423 577944 160873 048010 207879 285082 818345 533348 584778 223105 > 790 [i]