Best Known (231, 231+27, s)-Nets in Base 2
(231, 231+27, 40333)-Net over F2 — Constructive and digital
Digital (231, 258, 40333)-net over F2, using
- 23 times duplication [i] based on digital (228, 255, 40333)-net over F2, using
- net defined by OOA [i] based on linear OOA(2255, 40333, F2, 27, 27) (dual of [(40333, 27), 1088736, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2255, 524330, F2, 27) (dual of [524330, 524075, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2255, 524333, F2, 27) (dual of [524333, 524078, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2248, 524288, F2, 27) (dual of [524288, 524040, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2255, 524333, F2, 27) (dual of [524333, 524078, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2255, 524330, F2, 27) (dual of [524330, 524075, 28]-code), using
- net defined by OOA [i] based on linear OOA(2255, 40333, F2, 27, 27) (dual of [(40333, 27), 1088736, 28]-NRT-code), using
(231, 231+27, 74905)-Net over F2 — Digital
Digital (231, 258, 74905)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 74905, F2, 7, 27) (dual of [(74905, 7), 524077, 28]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2257, 74905, F2, 7, 27) (dual of [(74905, 7), 524078, 28]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2257, 524335, F2, 27) (dual of [524335, 524078, 28]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2255, 524333, F2, 27) (dual of [524333, 524078, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2248, 524288, F2, 27) (dual of [524288, 524040, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2255, 524333, F2, 27) (dual of [524333, 524078, 28]-code), using
- OOA 7-folding [i] based on linear OA(2257, 524335, F2, 27) (dual of [524335, 524078, 28]-code), using
- 21 times duplication [i] based on linear OOA(2257, 74905, F2, 7, 27) (dual of [(74905, 7), 524078, 28]-NRT-code), using
(231, 231+27, 5064493)-Net in Base 2 — Upper bound on s
There is no (231, 258, 5064494)-net in base 2, because
- 1 times m-reduction [i] would yield (231, 257, 5064494)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 231584 589721 040045 822777 995908 800669 248509 734238 646675 489570 218709 628363 386767 > 2257 [i]