Best Known (29, 29+9, s)-Nets in Base 4
(29, 29+9, 1028)-Net over F4 — Constructive and digital
Digital (29, 38, 1028)-net over F4, using
- 42 times duplication [i] based on digital (27, 36, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 9, 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, 9, 257)-net over F256, using
(29, 29+9, 2051)-Net over F4 — Digital
Digital (29, 38, 2051)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(438, 2051, F4, 2, 9) (dual of [(2051, 2), 4064, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(438, 4102, F4, 9) (dual of [4102, 4064, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(438, 4103, F4, 9) (dual of [4103, 4065, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(437, 4096, F4, 9) (dual of [4096, 4059, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(431, 4096, F4, 7) (dual of [4096, 4065, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(438, 4103, F4, 9) (dual of [4103, 4065, 10]-code), using
- OOA 2-folding [i] based on linear OA(438, 4102, F4, 9) (dual of [4102, 4064, 10]-code), using
(29, 29+9, 273515)-Net in Base 4 — Upper bound on s
There is no (29, 38, 273516)-net in base 4, because
- 1 times m-reduction [i] would yield (29, 37, 273516)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 18889 598128 167060 113746 > 437 [i]