Best Known (54−8, 54, s)-Nets in Base 5
(54−8, 54, 97670)-Net over F5 — Constructive and digital
Digital (46, 54, 97670)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 12)-net over F5, using
- digital (41, 49, 97658)-net over F5, using
- net defined by OOA [i] based on linear OOA(549, 97658, F5, 8, 8) (dual of [(97658, 8), 781215, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(549, 390632, F5, 8) (dual of [390632, 390583, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(549, 390633, F5, 8) (dual of [390633, 390584, 9]-code), using
- construction X 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(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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(549, 390633, F5, 8) (dual of [390633, 390584, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(549, 390632, F5, 8) (dual of [390632, 390583, 9]-code), using
- net defined by OOA [i] based on linear OOA(549, 97658, F5, 8, 8) (dual of [(97658, 8), 781215, 9]-NRT-code), using
(54−8, 54, 390654)-Net over F5 — Digital
Digital (46, 54, 390654)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(554, 390654, F5, 8) (dual of [390654, 390600, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(3) [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(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(55, 29, F5, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
(54−8, 54, large)-Net in Base 5 — Upper bound on s
There is no (46, 54, large)-net in base 5, because
- 6 times m-reduction [i] would yield (46, 48, large)-net in base 5, but