Best Known (42−12, 42, s)-Nets in Base 8
(42−12, 42, 684)-Net over F8 — Constructive and digital
Digital (30, 42, 684)-net over F8, using
- net defined by OOA [i] based on linear OOA(842, 684, F8, 12, 12) (dual of [(684, 12), 8166, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(842, 4104, F8, 12) (dual of [4104, 4062, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(842, 4105, F8, 12) (dual of [4105, 4063, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- 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(833, 4096, F8, 10) (dual of [4096, 4063, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(842, 4105, F8, 12) (dual of [4105, 4063, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(842, 4104, F8, 12) (dual of [4104, 4062, 13]-code), using
(42−12, 42, 3257)-Net over F8 — Digital
Digital (30, 42, 3257)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(842, 3257, F8, 12) (dual of [3257, 3215, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(842, 4105, F8, 12) (dual of [4105, 4063, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- 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(833, 4096, F8, 10) (dual of [4096, 4063, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(842, 4105, F8, 12) (dual of [4105, 4063, 13]-code), using
(42−12, 42, 896917)-Net in Base 8 — Upper bound on s
There is no (30, 42, 896918)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 85 071077 973501 428857 865958 520978 488968 > 842 [i]