Best Known (113, 113+13, s)-Nets in Base 4
(113, 113+13, 1398186)-Net over F4 — Constructive and digital
Digital (113, 126, 1398186)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (11, 17, 86)-net over F4, using
- net defined by OOA [i] based on linear OOA(417, 86, F4, 6, 6) (dual of [(86, 6), 499, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(417, 258, F4, 6) (dual of [258, 241, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(417, 260, F4, 6) (dual of [260, 243, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(417, 256, F4, 6) (dual of [256, 239, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(413, 256, F4, 5) (dual of [256, 243, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 4, F4, 0) (dual of [4, 4, 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
- discarding factors / shortening the dual code based on linear OA(417, 260, F4, 6) (dual of [260, 243, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(417, 258, F4, 6) (dual of [258, 241, 7]-code), using
- net defined by OOA [i] based on linear OOA(417, 86, F4, 6, 6) (dual of [(86, 6), 499, 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 (11, 17, 86)-net over F4, using
(113, 113+13, large)-Net over F4 — Digital
Digital (113, 126, large)-net over F4, using
- 42 times duplication [i] based on digital (111, 124, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4124, large, F4, 13) (dual of [large, large−124, 14]-code), using
- 15 times code embedding in larger space [i] 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]
- 15 times code embedding in larger space [i] based on linear OA(4109, large, F4, 13) (dual of [large, large−109, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4124, large, F4, 13) (dual of [large, large−124, 14]-code), using
(113, 113+13, large)-Net in Base 4 — Upper bound on s
There is no (113, 126, large)-net in base 4, because
- 11 times m-reduction [i] would yield (113, 115, large)-net in base 4, but