Best Known (68, 68+22, s)-Nets in Base 4
(68, 68+22, 1028)-Net over F4 — Constructive and digital
Digital (68, 90, 1028)-net over F4, using
- 42 times duplication [i] based on digital (66, 88, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 22, 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, 22, 257)-net over F256, using
(68, 68+22, 1149)-Net over F4 — Digital
Digital (68, 90, 1149)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(490, 1149, F4, 22) (dual of [1149, 1059, 23]-code), using
- 111 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 17 times 0, 1, 28 times 0, 1, 42 times 0) [i] based on linear OA(481, 1029, F4, 22) (dual of [1029, 948, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(481, 1024, F4, 22) (dual of [1024, 943, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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(21) ⊂ Ce(20) [i] based on
- 111 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 17 times 0, 1, 28 times 0, 1, 42 times 0) [i] based on linear OA(481, 1029, F4, 22) (dual of [1029, 948, 23]-code), using
(68, 68+22, 137978)-Net in Base 4 — Upper bound on s
There is no (68, 90, 137979)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 532572 598958 567727 433740 365168 629797 886178 138970 674868 > 490 [i]