Best Known (68, 68+12, s)-Nets in Base 5
(68, 68+12, 65115)-Net over F5 — Constructive and digital
Digital (68, 80, 65115)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (61, 73, 65105)-net over F5, using
- net defined by OOA [i] based on linear OOA(573, 65105, F5, 12, 12) (dual of [(65105, 12), 781187, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(573, 390630, F5, 12) (dual of [390630, 390557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(573, 390633, F5, 12) (dual of [390633, 390560, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(573, 390633, F5, 12) (dual of [390633, 390560, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(573, 390630, F5, 12) (dual of [390630, 390557, 13]-code), using
- net defined by OOA [i] based on linear OOA(573, 65105, F5, 12, 12) (dual of [(65105, 12), 781187, 13]-NRT-code), using
- digital (1, 7, 10)-net over F5, using
(68, 68+12, 376506)-Net over F5 — Digital
Digital (68, 80, 376506)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(580, 376506, F5, 12) (dual of [376506, 376426, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(580, 390635, F5, 12) (dual of [390635, 390555, 13]-code), using
- (u, u+v)-construction [i] based on
- linear OA(57, 10, F5, 6) (dual of [10, 3, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 11, F5, 6) (dual of [11, 4, 7]-code), using
- 1 times truncation [i] based on linear OA(58, 12, F5, 7) (dual of [12, 4, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 11, F5, 6) (dual of [11, 4, 7]-code), using
- 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(57, 10, F5, 6) (dual of [10, 3, 7]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(580, 390635, F5, 12) (dual of [390635, 390555, 13]-code), using
(68, 68+12, large)-Net in Base 5 — Upper bound on s
There is no (68, 80, large)-net in base 5, because
- 10 times m-reduction [i] would yield (68, 70, large)-net in base 5, but