Best Known (64−31, 64, s)-Nets in Base 256
(64−31, 64, 4369)-Net over F256 — Constructive and digital
Digital (33, 64, 4369)-net over F256, using
- 2563 times duplication [i] based on digital (30, 61, 4369)-net over F256, using
- net defined by OOA [i] based on linear OOA(25661, 4369, F256, 31, 31) (dual of [(4369, 31), 135378, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 15-folding and stacking with additional row [i] based on linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using
- net defined by OOA [i] based on linear OOA(25661, 4369, F256, 31, 31) (dual of [(4369, 31), 135378, 32]-NRT-code), using
(64−31, 64, 14924)-Net over F256 — Digital
Digital (33, 64, 14924)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25664, 14924, F256, 4, 31) (dual of [(14924, 4), 59632, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25664, 16387, F256, 4, 31) (dual of [(16387, 4), 65484, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25664, 65548, F256, 31) (dual of [65548, 65484, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(25661, 65537, F256, 31) (dual of [65537, 65476, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- OOA 4-folding [i] based on linear OA(25664, 65548, F256, 31) (dual of [65548, 65484, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(25664, 16387, F256, 4, 31) (dual of [(16387, 4), 65484, 32]-NRT-code), using
(64−31, 64, large)-Net in Base 256 — Upper bound on s
There is no (33, 64, large)-net in base 256, because
- 29 times m-reduction [i] would yield (33, 35, large)-net in base 256, but