Best Known (25−4, 25, s)-Nets in Base 5
(25−4, 25, 195316)-Net over F5 — Constructive and digital
Digital (21, 25, 195316)-net over F5, using
- net defined by OOA [i] based on linear OOA(525, 195316, F5, 4, 4) (dual of [(195316, 4), 781239, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(525, 195316, F5, 3, 4) (dual of [(195316, 3), 585923, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(525, 390632, F5, 4) (dual of [390632, 390607, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(525, 390633, F5, 4) (dual of [390633, 390608, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(525, 390625, F5, 4) (dual of [390625, 390600, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(517, 390625, F5, 3) (dual of [390625, 390608, 4]-code or 390625-cap in PG(16,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(525, 390633, F5, 4) (dual of [390633, 390608, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(525, 390632, F5, 4) (dual of [390632, 390607, 5]-code), using
- appending kth column [i] based on linear OOA(525, 195316, F5, 3, 4) (dual of [(195316, 3), 585923, 5]-NRT-code), using
(25−4, 25, 390633)-Net over F5 — Digital
Digital (21, 25, 390633)-net over F5, using
- net defined by OOA [i] based on linear OOA(525, 390633, F5, 4, 4) (dual of [(390633, 4), 1562507, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(525, 390633, F5, 3, 4) (dual of [(390633, 3), 1171874, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(525, 390633, F5, 4) (dual of [390633, 390608, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(525, 390625, F5, 4) (dual of [390625, 390600, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(517, 390625, F5, 3) (dual of [390625, 390608, 4]-code or 390625-cap in PG(16,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 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(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(525, 390633, F5, 4) (dual of [390633, 390608, 5]-code), using
- appending kth column [i] based on linear OOA(525, 390633, F5, 3, 4) (dual of [(390633, 3), 1171874, 5]-NRT-code), using
(25−4, 25, large)-Net in Base 5 — Upper bound on s
There is no (21, 25, large)-net in base 5, because
- 2 times m-reduction [i] would yield (21, 23, large)-net in base 5, but