Best Known (223−32, 223, s)-Nets in Base 2
(223−32, 223, 768)-Net over F2 — Constructive and digital
Digital (191, 223, 768)-net over F2, using
- 21 times duplication [i] based on digital (190, 222, 768)-net over F2, using
- trace code for nets [i] based on digital (5, 37, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 37, 128)-net over F64, using
(223−32, 223, 2579)-Net over F2 — Digital
Digital (191, 223, 2579)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2223, 2579, F2, 3, 32) (dual of [(2579, 3), 7514, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2223, 2748, F2, 3, 32) (dual of [(2748, 3), 8021, 33]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2222, 2748, F2, 3, 32) (dual of [(2748, 3), 8022, 33]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2222, 8244, F2, 32) (dual of [8244, 8022, 33]-code), using
- strength reduction [i] based on linear OA(2222, 8244, F2, 33) (dual of [8244, 8022, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [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(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(213, 52, F2, 5) (dual of [52, 39, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- strength reduction [i] based on linear OA(2222, 8244, F2, 33) (dual of [8244, 8022, 34]-code), using
- OOA 3-folding [i] based on linear OA(2222, 8244, F2, 32) (dual of [8244, 8022, 33]-code), using
- 21 times duplication [i] based on linear OOA(2222, 2748, F2, 3, 32) (dual of [(2748, 3), 8022, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2223, 2748, F2, 3, 32) (dual of [(2748, 3), 8021, 33]-NRT-code), using
(223−32, 223, 106671)-Net in Base 2 — Upper bound on s
There is no (191, 223, 106672)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 13 480869 378856 135805 279924 913515 807944 023086 691018 045736 671654 561298 > 2223 [i]