Best Known (131, 131+19, s)-Nets in Base 4
(131, 131+19, 116514)-Net over F4 — Constructive and digital
Digital (131, 150, 116514)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-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 (0, 9, 5)-net over F4, using
(131, 131+19, 524313)-Net over F4 — Digital
Digital (131, 150, 524313)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4150, 524313, F4, 2, 19) (dual of [(524313, 2), 1048476, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4150, 1048626, F4, 19) (dual of [1048626, 1048476, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(4141, 1048577, F4, 19) (dual of [1048577, 1048436, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(4101, 1048577, F4, 13) (dual of [1048577, 1048476, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(49, 49, F4, 5) (dual of [49, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(4150, 1048626, F4, 19) (dual of [1048626, 1048476, 20]-code), using
(131, 131+19, large)-Net in Base 4 — Upper bound on s
There is no (131, 150, large)-net in base 4, because
- 17 times m-reduction [i] would yield (131, 133, large)-net in base 4, but