Best Known (258−19, 258, s)-Nets in Base 2
(258−19, 258, 933093)-Net over F2 — Constructive and digital
Digital (239, 258, 933093)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (41, 50, 1027)-net over F2, using
- net defined by OOA [i] based on linear OOA(250, 1027, F2, 9, 9) (dual of [(1027, 9), 9193, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(250, 1027, F2, 8, 9) (dual of [(1027, 8), 8166, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(250, 4109, F2, 9) (dual of [4109, 4059, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(237, 4096, F2, 7) (dual of [4096, 4059, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(8) ⊂ Ce(6) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(250, 4109, F2, 9) (dual of [4109, 4059, 10]-code), using
- appending kth column [i] based on linear OOA(250, 1027, F2, 8, 9) (dual of [(1027, 8), 8166, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(250, 1027, F2, 9, 9) (dual of [(1027, 9), 9193, 10]-NRT-code), using
- digital (189, 208, 932066)-net over F2, using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- digital (41, 50, 1027)-net over F2, using
(258−19, 258, 1749835)-Net over F2 — Digital
Digital (239, 258, 1749835)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 1749835, F2, 4, 19) (dual of [(1749835, 4), 6999082, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2258, 2098178, F2, 4, 19) (dual of [(2098178, 4), 8392454, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2258, 4196356, F2, 2, 19) (dual of [(4196356, 2), 8392454, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(250, 2055, F2, 2, 9) (dual of [(2055, 2), 4060, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(250, 4110, F2, 9) (dual of [4110, 4060, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(237, 4096, F2, 7) (dual of [4096, 4059, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(250, 4110, F2, 9) (dual of [4110, 4060, 10]-code), using
- linear OOA(2208, 4194301, F2, 2, 19) (dual of [(4194301, 2), 8388394, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2208, 8388602, F2, 19) (dual of [8388602, 8388394, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 2-folding [i] based on linear OA(2208, 8388602, F2, 19) (dual of [8388602, 8388394, 20]-code), using
- linear OOA(250, 2055, F2, 2, 9) (dual of [(2055, 2), 4060, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA 2-folding [i] based on linear OOA(2258, 4196356, F2, 2, 19) (dual of [(4196356, 2), 8392454, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2258, 2098178, F2, 4, 19) (dual of [(2098178, 4), 8392454, 20]-NRT-code), using
(258−19, 258, large)-Net in Base 2 — Upper bound on s
There is no (239, 258, large)-net in base 2, because
- 17 times m-reduction [i] would yield (239, 241, large)-net in base 2, but