Best Known (33, 33+32, s)-Nets in Base 256
(33, 33+32, 4096)-Net over F256 — Constructive and digital
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
(33, 33+32, 13108)-Net over F256 — Digital
Digital (33, 65, 13108)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25665, 13108, F256, 5, 32) (dual of [(13108, 5), 65475, 33]-NRT-code), using
- 2561 times duplication [i] based on linear OOA(25664, 13108, F256, 5, 32) (dual of [(13108, 5), 65476, 33]-NRT-code), using
- OOA 5-folding [i] based on linear OA(25664, 65540, F256, 32) (dual of [65540, 65476, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(25664, 65541, F256, 32) (dual of [65541, 65477, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(29) [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(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(31) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(25664, 65541, F256, 32) (dual of [65541, 65477, 33]-code), using
- OOA 5-folding [i] based on linear OA(25664, 65540, F256, 32) (dual of [65540, 65476, 33]-code), using
- 2561 times duplication [i] based on linear OOA(25664, 13108, F256, 5, 32) (dual of [(13108, 5), 65476, 33]-NRT-code), using
(33, 33+32, large)-Net in Base 256 — Upper bound on s
There is no (33, 65, large)-net in base 256, because
- 30 times m-reduction [i] would yield (33, 35, large)-net in base 256, but