Best Known (63−14, 63, s)-Nets in Base 4
(63−14, 63, 1033)-Net over F4 — Constructive and digital
Digital (49, 63, 1033)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 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 (42, 56, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 14, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 14, 257)-net over F256, using
- digital (0, 7, 5)-net over F4, using
(63−14, 63, 2266)-Net over F4 — Digital
Digital (49, 63, 2266)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(463, 2266, F4, 14) (dual of [2266, 2203, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(463, 4105, F4, 14) (dual of [4105, 4042, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(461, 4096, F4, 14) (dual of [4096, 4035, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(455, 4096, F4, 13) (dual of [4096, 4041, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(449, 4096, F4, 11) (dual of [4096, 4047, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(463, 4105, F4, 14) (dual of [4105, 4042, 15]-code), using
(63−14, 63, 295344)-Net in Base 4 — Upper bound on s
There is no (49, 63, 295345)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 85 070797 390627 018546 736136 061933 714768 > 463 [i]