Best Known (51−14, 51, s)-Nets in Base 8
(51−14, 51, 586)-Net over F8 — Constructive and digital
Digital (37, 51, 586)-net over F8, using
- 81 times duplication [i] based on digital (36, 50, 586)-net over F8, using
- net defined by OOA [i] based on linear OOA(850, 586, F8, 14, 14) (dual of [(586, 14), 8154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(850, 4102, F8, 14) (dual of [4102, 4052, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(850, 4105, F8, 14) (dual of [4105, 4055, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(849, 4096, F8, 14) (dual of [4096, 4047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(841, 4096, F8, 12) (dual of [4096, 4055, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 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 Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(850, 4105, F8, 14) (dual of [4105, 4055, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(850, 4102, F8, 14) (dual of [4102, 4052, 15]-code), using
- net defined by OOA [i] based on linear OOA(850, 586, F8, 14, 14) (dual of [(586, 14), 8154, 15]-NRT-code), using
(51−14, 51, 4109)-Net over F8 — Digital
Digital (37, 51, 4109)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(851, 4109, F8, 14) (dual of [4109, 4058, 15]-code), using
- 7 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0) [i] based on linear OA(849, 4100, F8, 14) (dual of [4100, 4051, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(849, 4096, F8, 14) (dual of [4096, 4047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 7 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0) [i] based on linear OA(849, 4100, F8, 14) (dual of [4100, 4051, 15]-code), using
(51−14, 51, 1834320)-Net in Base 8 — Upper bound on s
There is no (37, 51, 1834321)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 11417 992130 085948 527952 121730 262797 810433 267664 > 851 [i]