Best Known (38, 38+6, s)-Nets in Base 4
(38, 38+6, 349533)-Net over F4 — Constructive and digital
Digital (38, 44, 349533)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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 (35, 41, 349528)-net over F4, using
- net defined by OOA [i] based on linear OOA(441, 349528, F4, 6, 6) (dual of [(349528, 6), 2097127, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(441, 1048584, F4, 6) (dual of [1048584, 1048543, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(441, 1048586, F4, 6) (dual of [1048586, 1048545, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(441, 1048576, F4, 6) (dual of [1048576, 1048535, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(441, 1048586, F4, 6) (dual of [1048586, 1048545, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(441, 1048584, F4, 6) (dual of [1048584, 1048543, 7]-code), using
- net defined by OOA [i] based on linear OOA(441, 349528, F4, 6, 6) (dual of [(349528, 6), 2097127, 7]-NRT-code), using
- digital (0, 3, 5)-net over F4, using
(38, 38+6, 1048597)-Net over F4 — Digital
Digital (38, 44, 1048597)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(444, 1048597, F4, 6) (dual of [1048597, 1048553, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(441, 1048576, F4, 6) (dual of [1048576, 1048535, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(421, 1048576, F4, 3) (dual of [1048576, 1048555, 4]-code or 1048576-cap in PG(20,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
(38, 38+6, large)-Net in Base 4 — Upper bound on s
There is no (38, 44, large)-net in base 4, because
- 4 times m-reduction [i] would yield (38, 40, large)-net in base 4, but