Best Known (108, 133, s)-Nets in Base 2
(108, 133, 260)-Net over F2 — Constructive and digital
Digital (108, 133, 260)-net over F2, using
- 3 times m-reduction [i] based on digital (108, 136, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 34, 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, 34, 65)-net over F16, using
(108, 133, 604)-Net over F2 — Digital
Digital (108, 133, 604)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2133, 604, F2, 3, 25) (dual of [(604, 3), 1679, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2133, 683, F2, 3, 25) (dual of [(683, 3), 1916, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2133, 2049, F2, 25) (dual of [2049, 1916, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(2133, 2049, F2, 25) (dual of [2049, 1916, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(2133, 683, F2, 3, 25) (dual of [(683, 3), 1916, 26]-NRT-code), using
(108, 133, 10814)-Net in Base 2 — Upper bound on s
There is no (108, 133, 10815)-net in base 2, because
- 1 times m-reduction [i] would yield (108, 132, 10815)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5450 060040 128817 824765 501153 695252 439853 > 2132 [i]