Best Known (15−7, 15, s)-Nets in Base 256
(15−7, 15, 21847)-Net over F256 — Constructive and digital
Digital (8, 15, 21847)-net over F256, using
- 2561 times duplication [i] based on digital (7, 14, 21847)-net over F256, using
- net defined by OOA [i] based on linear OOA(25614, 21847, F256, 7, 7) (dual of [(21847, 7), 152915, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25614, 21847, F256, 6, 7) (dual of [(21847, 6), 131068, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(25613, 65537, F256, 7) (dual of [65537, 65524, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(2569, 65537, F256, 5) (dual of [65537, 65528, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [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 C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- appending kth column [i] based on linear OOA(25614, 21847, F256, 6, 7) (dual of [(21847, 6), 131068, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25614, 21847, F256, 7, 7) (dual of [(21847, 7), 152915, 8]-NRT-code), using
(15−7, 15, 56540)-Net over F256 — Digital
Digital (8, 15, 56540)-net over F256, using
- net defined by OOA [i] based on linear OOA(25615, 56540, F256, 7, 7) (dual of [(56540, 7), 395765, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25615, 56540, F256, 6, 7) (dual of [(56540, 6), 339225, 8]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25615, 56540, F256, 7) (dual of [56540, 56525, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(25615, 65540, F256, 7) (dual of [65540, 65525, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- linear OA(25613, 65537, F256, 7) (dual of [65537, 65524, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(25612, 65537, F256, 4) (dual of [65537, 65525, 5]-code), using the narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [1,3], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- linear OA(2562, 3, F256, 2) (dual of [3, 1, 3]-code or 3-arc in PG(1,256)), using
- dual of repetition code with length 3 [i]
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25615, 65540, F256, 7) (dual of [65540, 65525, 8]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25615, 56540, F256, 7) (dual of [56540, 56525, 8]-code), using
- appending kth column [i] based on linear OOA(25615, 56540, F256, 6, 7) (dual of [(56540, 6), 339225, 8]-NRT-code), using
(15−7, 15, large)-Net in Base 256 — Upper bound on s
There is no (8, 15, large)-net in base 256, because
- 5 times m-reduction [i] would yield (8, 10, large)-net in base 256, but