Best Known (178, 178+32, s)-Nets in Base 2
(178, 178+32, 624)-Net over F2 — Constructive and digital
Digital (178, 210, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 35, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
(178, 178+32, 2051)-Net over F2 — Digital
Digital (178, 210, 2051)-net over F2, using
- 21 times duplication [i] based on digital (177, 209, 2051)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2209, 2051, F2, 4, 32) (dual of [(2051, 4), 7995, 33]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2209, 8204, F2, 32) (dual of [8204, 7995, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(2209, 8205, F2, 32) (dual of [8205, 7996, 33]-code), using
- 1 times truncation [i] based on linear OA(2210, 8206, F2, 33) (dual of [8206, 7996, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2209, 8192, F2, 33) (dual of [8192, 7983, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- 1 times truncation [i] based on linear OA(2210, 8206, F2, 33) (dual of [8206, 7996, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2209, 8205, F2, 32) (dual of [8205, 7996, 33]-code), using
- OOA 4-folding [i] based on linear OA(2209, 8204, F2, 32) (dual of [8204, 7995, 33]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2209, 2051, F2, 4, 32) (dual of [(2051, 4), 7995, 33]-NRT-code), using
(178, 178+32, 60728)-Net in Base 2 — Upper bound on s
There is no (178, 210, 60729)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1645 911198 595739 966777 630340 259507 728114 109539 523298 524166 727385 > 2210 [i]