Best Known (89−19, 89, s)-Nets in Base 4
(89−19, 89, 1043)-Net over F4 — Constructive and digital
Digital (70, 89, 1043)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- digital (57, 76, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 19, 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, 19, 257)-net over F256, using
- digital (4, 13, 15)-net over F4, using
(89−19, 89, 3116)-Net over F4 — Digital
Digital (70, 89, 3116)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(489, 3116, F4, 19) (dual of [3116, 3027, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 4114, F4, 19) (dual of [4114, 4025, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(485, 4096, F4, 19) (dual of [4096, 4011, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(473, 4096, F4, 17) (dual of [4096, 4023, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(467, 4096, F4, 15) (dual of [4096, 4029, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(489, 4114, F4, 19) (dual of [4114, 4025, 20]-code), using
(89−19, 89, 1065212)-Net in Base 4 — Upper bound on s
There is no (70, 89, 1065213)-net in base 4, because
- 1 times m-reduction [i] would yield (70, 88, 1065213)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 95781 595442 678419 539707 367265 207068 276228 017149 851040 > 488 [i]