Best Known (75, 75+31, s)-Nets in Base 7
(75, 75+31, 688)-Net over F7 — Constructive and digital
Digital (75, 106, 688)-net over F7, using
- 2 times m-reduction [i] based on digital (75, 108, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 54, 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, 54, 344)-net over F49, using
(75, 75+31, 2215)-Net over F7 — Digital
Digital (75, 106, 2215)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7106, 2215, F7, 31) (dual of [2215, 2109, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(7106, 2411, F7, 31) (dual of [2411, 2305, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- linear OA(7105, 2402, F7, 31) (dual of [2402, 2297, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(797, 2402, F7, 29) (dual of [2402, 2305, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(7106, 2411, F7, 31) (dual of [2411, 2305, 32]-code), using
(75, 75+31, 881651)-Net in Base 7 — Upper bound on s
There is no (75, 106, 881652)-net in base 7, because
- 1 times m-reduction [i] would yield (75, 105, 881652)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 54362 233773 227255 760164 488028 641085 189223 455238 802317 617891 256866 500738 521373 466709 097633 > 7105 [i]