Best Known (36, 36+31, s)-Nets in Base 256
(36, 36+31, 4370)-Net over F256 — Constructive and digital
Digital (36, 67, 4370)-net over F256, using
- 2561 times duplication [i] based on digital (35, 66, 4370)-net over F256, using
- net defined by OOA [i] based on linear OOA(25666, 4370, F256, 31, 31) (dual of [(4370, 31), 135404, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25666, 65551, F256, 31) (dual of [65551, 65485, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [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(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2565, 17, F256, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- Reed–Solomon code RS(251,256) [i]
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25666, 65551, F256, 31) (dual of [65551, 65485, 32]-code), using
- net defined by OOA [i] based on linear OOA(25666, 4370, F256, 31, 31) (dual of [(4370, 31), 135404, 32]-NRT-code), using
(36, 36+31, 21852)-Net over F256 — Digital
Digital (36, 67, 21852)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25667, 21852, F256, 3, 31) (dual of [(21852, 3), 65489, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25667, 65556, F256, 31) (dual of [65556, 65489, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(23) [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]
- linear OA(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2566, 20, F256, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,256)), using
- discarding factors / shortening the dual code based on linear OA(2566, 256, F256, 6) (dual of [256, 250, 7]-code or 256-arc in PG(5,256)), using
- Reed–Solomon code RS(250,256) [i]
- discarding factors / shortening the dual code based on linear OA(2566, 256, F256, 6) (dual of [256, 250, 7]-code or 256-arc in PG(5,256)), using
- construction X applied to Ce(30) ⊂ Ce(23) [i] based on
- OOA 3-folding [i] based on linear OA(25667, 65556, F256, 31) (dual of [65556, 65489, 32]-code), using
(36, 36+31, large)-Net in Base 256 — Upper bound on s
There is no (36, 67, large)-net in base 256, because
- 29 times m-reduction [i] would yield (36, 38, large)-net in base 256, but