Best Known (67−32, 67, s)-Nets in Base 256
(67−32, 67, 4096)-Net over F256 — Constructive and digital
Digital (35, 67, 4096)-net over F256, using
- 2562 times duplication [i] based on digital (33, 65, 4096)-net over F256, using
- t-expansion [i] based on digital (32, 65, 4096)-net over F256, using
- net defined by OOA [i] based on linear OOA(25665, 4096, F256, 33, 33) (dual of [(4096, 33), 135103, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
- net defined by OOA [i] based on linear OOA(25665, 4096, F256, 33, 33) (dual of [(4096, 33), 135103, 34]-NRT-code), using
- t-expansion [i] based on digital (32, 65, 4096)-net over F256, using
(67−32, 67, 16387)-Net over F256 — Digital
Digital (35, 67, 16387)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25667, 16387, F256, 4, 32) (dual of [(16387, 4), 65481, 33]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25667, 65548, F256, 32) (dual of [65548, 65481, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(25667, 65550, F256, 32) (dual of [65550, 65483, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(26) [i] based on
- linear OA(25663, 65536, F256, 32) (dual of [65536, 65473, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(25653, 65536, F256, 27) (dual of [65536, 65483, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(31) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(25667, 65550, F256, 32) (dual of [65550, 65483, 33]-code), using
- OOA 4-folding [i] based on linear OA(25667, 65548, F256, 32) (dual of [65548, 65481, 33]-code), using
(67−32, 67, large)-Net in Base 256 — Upper bound on s
There is no (35, 67, large)-net in base 256, because
- 30 times m-reduction [i] would yield (35, 37, large)-net in base 256, but