Best Known (42, 42+8, s)-Nets in Base 5
(42, 42+8, 97660)-Net over F5 — Constructive and digital
Digital (42, 50, 97660)-net over F5, using
- net defined by OOA [i] based on linear OOA(550, 97660, F5, 8, 8) (dual of [(97660, 8), 781230, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(550, 390640, F5, 8) (dual of [390640, 390590, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(550, 390642, F5, 8) (dual of [390642, 390592, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(51, 17, F5, 1) (dual of [17, 16, 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 X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(550, 390642, F5, 8) (dual of [390642, 390592, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(550, 390640, F5, 8) (dual of [390640, 390590, 9]-code), using
(42, 42+8, 382308)-Net over F5 — Digital
Digital (42, 50, 382308)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(550, 382308, F5, 8) (dual of [382308, 382258, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(550, 390635, F5, 8) (dual of [390635, 390585, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(550, 390635, F5, 8) (dual of [390635, 390585, 9]-code), using
(42, 42+8, large)-Net in Base 5 — Upper bound on s
There is no (42, 50, large)-net in base 5, because
- 6 times m-reduction [i] would yield (42, 44, large)-net in base 5, but