Best Known (42, 42+14, s)-Nets in Base 5
(42, 42+14, 447)-Net over F5 — Constructive and digital
Digital (42, 56, 447)-net over F5, using
- net defined by OOA [i] based on linear OOA(556, 447, F5, 14, 14) (dual of [(447, 14), 6202, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(556, 3129, F5, 14) (dual of [3129, 3073, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(556, 3130, F5, 14) (dual of [3130, 3074, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(556, 3129, F5, 14) (dual of [3129, 3073, 15]-code), using
(42, 42+14, 2105)-Net over F5 — Digital
Digital (42, 56, 2105)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(556, 2105, F5, 14) (dual of [2105, 2049, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using
(42, 42+14, 330074)-Net in Base 5 — Upper bound on s
There is no (42, 56, 330075)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1387 783036 137639 441759 699836 604975 879061 > 556 [i]