Best Known (20, 37, s)-Nets in Base 256
(20, 37, 8193)-Net over F256 — Constructive and digital
Digital (20, 37, 8193)-net over F256, using
- 2561 times duplication [i] based on digital (19, 36, 8193)-net over F256, using
- net defined by OOA [i] based on linear OOA(25636, 8193, F256, 17, 17) (dual of [(8193, 17), 139245, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25636, 65545, F256, 17) (dual of [65545, 65509, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25636, 65548, F256, 17) (dual of [65548, 65512, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25636, 65548, F256, 17) (dual of [65548, 65512, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25636, 65545, F256, 17) (dual of [65545, 65509, 18]-code), using
- net defined by OOA [i] based on linear OOA(25636, 8193, F256, 17, 17) (dual of [(8193, 17), 139245, 18]-NRT-code), using
(20, 37, 24855)-Net over F256 — Digital
Digital (20, 37, 24855)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25637, 24855, F256, 2, 17) (dual of [(24855, 2), 49673, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25637, 32775, F256, 2, 17) (dual of [(32775, 2), 65513, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25637, 65550, F256, 17) (dual of [65550, 65513, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(25633, 65536, F256, 17) (dual of [65536, 65503, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(25637, 65550, F256, 17) (dual of [65550, 65513, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(25637, 32775, F256, 2, 17) (dual of [(32775, 2), 65513, 18]-NRT-code), using
(20, 37, large)-Net in Base 256 — Upper bound on s
There is no (20, 37, large)-net in base 256, because
- 15 times m-reduction [i] would yield (20, 22, large)-net in base 256, but