Best Known (107−31, 107, s)-Nets in Base 7
(107−31, 107, 688)-Net over F7 — Constructive and digital
Digital (76, 107, 688)-net over F7, using
- 3 times m-reduction [i] based on digital (76, 110, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 55, 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, 55, 344)-net over F49, using
(107−31, 107, 2370)-Net over F7 — Digital
Digital (76, 107, 2370)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7107, 2370, F7, 31) (dual of [2370, 2263, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(7107, 2412, F7, 31) (dual of [2412, 2305, 32]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(7106, 2411, F7, 31) (dual of [2411, 2305, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(7107, 2412, F7, 31) (dual of [2412, 2305, 32]-code), using
(107−31, 107, 1003777)-Net in Base 7 — Upper bound on s
There is no (76, 107, 1003778)-net in base 7, because
- 1 times m-reduction [i] would yield (76, 106, 1003778)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 380536 125638 130980 476560 769507 441029 265636 843752 560541 517731 719145 928600 458246 980255 282353 > 7106 [i]