Best Known (43−7, 43, s)-Nets in Base 5
(43−7, 43, 130211)-Net over F5 — Constructive and digital
Digital (36, 43, 130211)-net over F5, using
- 51 times duplication [i] based on digital (35, 42, 130211)-net over F5, using
- net defined by OOA [i] based on linear OOA(542, 130211, F5, 7, 7) (dual of [(130211, 7), 911435, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(542, 390634, F5, 7) (dual of [390634, 390592, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(542, 390635, F5, 7) (dual of [390635, 390593, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(541, 390625, F5, 7) (dual of [390625, 390584, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(533, 390625, F5, 6) (dual of [390625, 390592, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(59, 10, F5, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,5)), using
- dual of repetition code with length 10 [i]
- linear OA(51, 10, F5, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(542, 390635, F5, 7) (dual of [390635, 390593, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(542, 390634, F5, 7) (dual of [390634, 390592, 8]-code), using
- net defined by OOA [i] based on linear OOA(542, 130211, F5, 7, 7) (dual of [(130211, 7), 911435, 8]-NRT-code), using
(43−7, 43, 390637)-Net over F5 — Digital
Digital (36, 43, 390637)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(543, 390637, F5, 7) (dual of [390637, 390594, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(542, 390635, F5, 7) (dual of [390635, 390593, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(541, 390625, F5, 7) (dual of [390625, 390584, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(533, 390625, F5, 6) (dual of [390625, 390592, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(59, 10, F5, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,5)), using
- dual of repetition code with length 10 [i]
- linear OA(51, 10, F5, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(542, 390636, F5, 6) (dual of [390636, 390594, 7]-code), using Gilbert–Varšamov bound and bm = 542 > Vbs−1(k−1) = 77618 410535 183852 208369 382749 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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
- linear OA(542, 390635, F5, 7) (dual of [390635, 390593, 8]-code), using
- construction X with Varšamov bound [i] based on
(43−7, 43, large)-Net in Base 5 — Upper bound on s
There is no (36, 43, large)-net in base 5, because
- 5 times m-reduction [i] would yield (36, 38, large)-net in base 5, but