Best Known (153−19, 153, s)-Nets in Base 4
(153−19, 153, 116523)-Net over F4 — Constructive and digital
Digital (134, 153, 116523)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (122, 141, 116509)-net over F4, using
- net defined by OOA [i] based on linear OOA(4141, 116509, F4, 19, 19) (dual of [(116509, 19), 2213530, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4141, 1048582, F4, 19) (dual of [1048582, 1048441, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4141, 1048586, F4, 19) (dual of [1048586, 1048445, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4141, 1048586, F4, 19) (dual of [1048586, 1048445, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4141, 1048582, F4, 19) (dual of [1048582, 1048441, 20]-code), using
- net defined by OOA [i] based on linear OOA(4141, 116509, F4, 19, 19) (dual of [(116509, 19), 2213530, 20]-NRT-code), using
- digital (3, 12, 14)-net over F4, using
(153−19, 153, 578014)-Net over F4 — Digital
Digital (134, 153, 578014)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4153, 578014, F4, 19) (dual of [578014, 577861, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4153, 1048638, F4, 19) (dual of [1048638, 1048485, 20]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4151, 1048636, F4, 19) (dual of [1048636, 1048485, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4151, 1048636, F4, 19) (dual of [1048636, 1048485, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4153, 1048638, F4, 19) (dual of [1048638, 1048485, 20]-code), using
(153−19, 153, large)-Net in Base 4 — Upper bound on s
There is no (134, 153, large)-net in base 4, because
- 17 times m-reduction [i] would yield (134, 136, large)-net in base 4, but