Best Known (142−13, 142, s)-Nets in Base 4
(142−13, 142, 1419948)-Net over F4 — Constructive and digital
Digital (129, 142, 1419948)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (27, 33, 21848)-net over F4, using
- net defined by OOA [i] based on linear OOA(433, 21848, F4, 6, 6) (dual of [(21848, 6), 131055, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(433, 65544, F4, 6) (dual of [65544, 65511, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(433, 65544, F4, 6) (dual of [65544, 65511, 7]-code), using
- net defined by OOA [i] based on linear OOA(433, 21848, F4, 6, 6) (dual of [(21848, 6), 131055, 7]-NRT-code), using
- digital (96, 109, 1398100)-net over F4, using
- net defined by OOA [i] based on linear OOA(4109, 1398100, F4, 13, 13) (dual of [(1398100, 13), 18175191, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4109, 8388601, F4, 13) (dual of [8388601, 8388492, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4109, 8388601, F4, 13) (dual of [8388601, 8388492, 14]-code), using
- net defined by OOA [i] based on linear OOA(4109, 1398100, F4, 13, 13) (dual of [(1398100, 13), 18175191, 14]-NRT-code), using
- digital (27, 33, 21848)-net over F4, using
(142−13, 142, large)-Net over F4 — Digital
Digital (129, 142, large)-net over F4, using
- 47 times duplication [i] based on digital (122, 135, large)-net over F4, using
- t-expansion [i] based on digital (121, 135, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4135, large, F4, 14) (dual of [large, large−135, 15]-code), using
- 14 times code embedding in larger space [i] based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 14 times code embedding in larger space [i] based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4135, large, F4, 14) (dual of [large, large−135, 15]-code), using
- t-expansion [i] based on digital (121, 135, large)-net over F4, using
(142−13, 142, large)-Net in Base 4 — Upper bound on s
There is no (129, 142, large)-net in base 4, because
- 11 times m-reduction [i] would yield (129, 131, large)-net in base 4, but