Best Known (81, 81+19, s)-Nets in Base 8
(81, 81+19, 29128)-Net over F8 — Constructive and digital
Digital (81, 100, 29128)-net over F8, using
- 82 times duplication [i] based on digital (79, 98, 29128)-net over F8, using
- net defined by OOA [i] based on linear OOA(898, 29128, F8, 19, 19) (dual of [(29128, 19), 553334, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(898, 262153, F8, 19) (dual of [262153, 262055, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(898, 262158, F8, 19) (dual of [262158, 262060, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(897, 262145, F8, 19) (dual of [262145, 262048, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(885, 262145, F8, 17) (dual of [262145, 262060, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(81, 13, F8, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(898, 262158, F8, 19) (dual of [262158, 262060, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(898, 262153, F8, 19) (dual of [262153, 262055, 20]-code), using
- net defined by OOA [i] based on linear OOA(898, 29128, F8, 19, 19) (dual of [(29128, 19), 553334, 20]-NRT-code), using
(81, 81+19, 186207)-Net over F8 — Digital
Digital (81, 100, 186207)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8100, 186207, F8, 19) (dual of [186207, 186107, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8100, 262160, F8, 19) (dual of [262160, 262060, 20]-code), using
- 2 times code embedding in larger space [i] based on linear OA(898, 262158, F8, 19) (dual of [262158, 262060, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(897, 262145, F8, 19) (dual of [262145, 262048, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(885, 262145, F8, 17) (dual of [262145, 262060, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(81, 13, F8, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(898, 262158, F8, 19) (dual of [262158, 262060, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8100, 262160, F8, 19) (dual of [262160, 262060, 20]-code), using
(81, 81+19, large)-Net in Base 8 — Upper bound on s
There is no (81, 100, large)-net in base 8, because
- 17 times m-reduction [i] would yield (81, 83, large)-net in base 8, but