Best Known (69, 69+23, s)-Nets in Base 7
(69, 69+23, 688)-Net over F7 — Constructive and digital
Digital (69, 92, 688)-net over F7, using
- 4 times m-reduction [i] based on digital (69, 96, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 48, 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, 48, 344)-net over F49, using
(69, 69+23, 5211)-Net over F7 — Digital
Digital (69, 92, 5211)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(792, 5211, F7, 23) (dual of [5211, 5119, 24]-code), using
- 403 step Varšamov–Edel lengthening with (ri) = (1, 104 times 0, 1, 297 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
- 403 step Varšamov–Edel lengthening with (ri) = (1, 104 times 0, 1, 297 times 0) [i] based on linear OA(790, 4806, F7, 23) (dual of [4806, 4716, 24]-code), using
(69, 69+23, 8019146)-Net in Base 7 — Upper bound on s
There is no (69, 92, 8019147)-net in base 7, because
- 1 times m-reduction [i] would yield (69, 91, 8019147)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 80153 414570 251208 067801 825538 214901 363464 348238 521419 437620 052557 596654 723903 > 791 [i]