Best Known (38−10, 38, s)-Nets in Base 4
(38−10, 38, 514)-Net over F4 — Constructive and digital
Digital (28, 38, 514)-net over F4, using
- trace code for nets [i] based on digital (9, 19, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(9,256) in PG(18,16)) for nets [i] based on digital (0, 10, 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
- base reduction for projective spaces (embedding PG(9,256) in PG(18,16)) for nets [i] based on digital (0, 10, 257)-net over F256, using
(38−10, 38, 758)-Net over F4 — Digital
Digital (28, 38, 758)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(438, 758, F4, 10) (dual of [758, 720, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(438, 1032, F4, 10) (dual of [1032, 994, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(8) ⊂ Ce(6) [i] based on
- linear OA(436, 1024, F4, 10) (dual of [1024, 988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(431, 1024, F4, 9) (dual of [1024, 993, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(426, 1024, F4, 7) (dual of [1024, 998, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 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(9) ⊂ Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(438, 1032, F4, 10) (dual of [1032, 994, 11]-code), using
(38−10, 38, 32683)-Net in Base 4 — Upper bound on s
There is no (28, 38, 32684)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 75569 164282 877429 735482 > 438 [i]