Best Known (114−21, 114, s)-Nets in Base 8
(114−21, 114, 26217)-Net over F8 — Constructive and digital
Digital (93, 114, 26217)-net over F8, using
- 81 times duplication [i] based on digital (92, 113, 26217)-net over F8, using
- net defined by OOA [i] based on linear OOA(8113, 26217, F8, 21, 21) (dual of [(26217, 21), 550444, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8113, 262171, F8, 21) (dual of [262171, 262058, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, 262173, F8, 21) (dual of [262173, 262060, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(8109, 262145, F8, 21) (dual of [262145, 262036, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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(84, 28, F8, 3) (dual of [28, 24, 4]-code or 28-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(8113, 262173, F8, 21) (dual of [262173, 262060, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8113, 262171, F8, 21) (dual of [262171, 262058, 22]-code), using
- net defined by OOA [i] based on linear OOA(8113, 26217, F8, 21, 21) (dual of [(26217, 21), 550444, 22]-NRT-code), using
(114−21, 114, 262175)-Net over F8 — Digital
Digital (93, 114, 262175)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8114, 262175, F8, 21) (dual of [262175, 262061, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8113, 262173, F8, 21) (dual of [262173, 262060, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(8109, 262145, F8, 21) (dual of [262145, 262036, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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(84, 28, F8, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,8)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(8113, 262174, F8, 20) (dual of [262174, 262061, 21]-code), using Gilbert–Varšamov bound and bm = 8113 > Vbs−1(k−1) = 840737 993026 812655 501009 222403 482943 967054 533822 865466 612498 549977 968318 516672 439606 915542 744250 309140 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(8113, 262173, F8, 21) (dual of [262173, 262060, 22]-code), using
- construction X with Varšamov bound [i] based on
(114−21, 114, large)-Net in Base 8 — Upper bound on s
There is no (93, 114, large)-net in base 8, because
- 19 times m-reduction [i] would yield (93, 95, large)-net in base 8, but