Best Known (112, 112+21, s)-Nets in Base 8
(112, 112+21, 209718)-Net over F8 — Constructive and digital
Digital (112, 133, 209718)-net over F8, using
- 82 times duplication [i] based on digital (110, 131, 209718)-net over F8, using
- net defined by OOA [i] based on linear OOA(8131, 209718, F8, 21, 21) (dual of [(209718, 21), 4403947, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8131, 2097181, F8, 21) (dual of [2097181, 2097050, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8131, 2097185, F8, 21) (dual of [2097185, 2097054, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(8127, 2097153, F8, 21) (dual of [2097153, 2097026, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(899, 2097153, F8, 17) (dual of [2097153, 2097054, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(84, 32, F8, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,8)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8131, 2097185, F8, 21) (dual of [2097185, 2097054, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8131, 2097181, F8, 21) (dual of [2097181, 2097050, 22]-code), using
- net defined by OOA [i] based on linear OOA(8131, 209718, F8, 21, 21) (dual of [(209718, 21), 4403947, 22]-NRT-code), using
(112, 112+21, 2097188)-Net over F8 — Digital
Digital (112, 133, 2097188)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8133, 2097188, F8, 21) (dual of [2097188, 2097055, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8131, 2097185, F8, 21) (dual of [2097185, 2097054, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(8127, 2097153, F8, 21) (dual of [2097153, 2097026, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(899, 2097153, F8, 17) (dual of [2097153, 2097054, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(84, 32, F8, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,8)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(8131, 2097186, F8, 19) (dual of [2097186, 2097055, 20]-code), using Gilbert–Varšamov bound and bm = 8131 > Vbs−1(k−1) = 156622 415883 295212 979985 346426 096320 078843 835802 306984 247222 152339 037234 828478 512133 581540 768823 386907 724914 306664 [i]
- linear OA(81, 2, F8, 1) (dual of [2, 1, 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
- linear OA(8131, 2097185, F8, 21) (dual of [2097185, 2097054, 22]-code), using
- construction X with Varšamov bound [i] based on
(112, 112+21, large)-Net in Base 8 — Upper bound on s
There is no (112, 133, large)-net in base 8, because
- 19 times m-reduction [i] would yield (112, 114, large)-net in base 8, but