Best Known (77, 90, s)-Nets in Base 5
(77, 90, 65132)-Net over F5 — Constructive and digital
Digital (77, 90, 65132)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 27)-net over F5, using
- digital (68, 81, 65105)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 65105, F5, 13, 13) (dual of [(65105, 13), 846284, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(581, 390631, F5, 13) (dual of [390631, 390550, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 390633, F5, 13) (dual of [390633, 390552, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(581, 390633, F5, 13) (dual of [390633, 390552, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(581, 390631, F5, 13) (dual of [390631, 390550, 14]-code), using
- net defined by OOA [i] based on linear OOA(581, 65105, F5, 13, 13) (dual of [(65105, 13), 846284, 14]-NRT-code), using
(77, 90, 390674)-Net over F5 — Digital
Digital (77, 90, 390674)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(590, 390674, F5, 13) (dual of [390674, 390584, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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, 49, F5, 5) (dual of [49, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
(77, 90, large)-Net in Base 5 — Upper bound on s
There is no (77, 90, large)-net in base 5, because
- 11 times m-reduction [i] would yield (77, 79, large)-net in base 5, but