Best Known (121, 121+12, s)-Nets in Base 4
(121, 121+12, 1399467)-Net over F4 — Constructive and digital
Digital (121, 133, 1399467)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (19, 25, 1367)-net over F4, using
- net defined by OOA [i] based on linear OOA(425, 1367, F4, 6, 6) (dual of [(1367, 6), 8177, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(425, 4101, F4, 6) (dual of [4101, 4076, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(425, 4102, F4, 6) (dual of [4102, 4077, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(425, 4096, F4, 6) (dual of [4096, 4071, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 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(425, 4102, F4, 6) (dual of [4102, 4077, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(425, 4101, F4, 6) (dual of [4101, 4076, 7]-code), using
- net defined by OOA [i] based on linear OOA(425, 1367, F4, 6, 6) (dual of [(1367, 6), 8177, 7]-NRT-code), using
- digital (96, 108, 1398100)-net over F4, using
- net defined by OOA [i] based on linear OOA(4108, 1398100, F4, 12, 12) (dual of [(1398100, 12), 16777092, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4108, 8388600, F4, 12) (dual of [8388600, 8388492, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(4108, large, F4, 12) (dual of [large, large−108, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(4108, large, F4, 12) (dual of [large, large−108, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(4108, 8388600, F4, 12) (dual of [8388600, 8388492, 13]-code), using
- net defined by OOA [i] based on linear OOA(4108, 1398100, F4, 12, 12) (dual of [(1398100, 12), 16777092, 13]-NRT-code), using
- digital (19, 25, 1367)-net over F4, using
(121, 121+12, large)-Net over F4 — Digital
Digital (121, 133, large)-net over F4, using
- 2 times m-reduction [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
(121, 121+12, large)-Net in Base 4 — Upper bound on s
There is no (121, 133, large)-net in base 4, because
- 10 times m-reduction [i] would yield (121, 123, large)-net in base 4, but