Best Known (117−21, 117, s)-Nets in Base 2
(117−21, 117, 260)-Net over F2 — Constructive and digital
Digital (96, 117, 260)-net over F2, using
- 3 times m-reduction [i] based on digital (96, 120, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 30, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 30, 65)-net over F16, using
(117−21, 117, 692)-Net over F2 — Digital
Digital (96, 117, 692)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2117, 692, F2, 3, 21) (dual of [(692, 3), 1959, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2117, 2076, F2, 21) (dual of [2076, 1959, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2117, 2077, F2, 21) (dual of [2077, 1960, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2111, 2049, F2, 21) (dual of [2049, 1938, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(289, 2049, F2, 17) (dual of [2049, 1960, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2117, 2077, F2, 21) (dual of [2077, 1960, 22]-code), using
- OOA 3-folding [i] based on linear OA(2117, 2076, F2, 21) (dual of [2076, 1959, 22]-code), using
(117−21, 117, 14043)-Net in Base 2 — Upper bound on s
There is no (96, 117, 14044)-net in base 2, because
- 1 times m-reduction [i] would yield (96, 116, 14044)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 83104 540445 105009 829405 191532 461029 > 2116 [i]