Best Known (42, 42+6, s)-Nets in Base 4
(42, 42+6, 1398110)-Net over F4 — Constructive and digital
Digital (42, 48, 1398110)-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 (39, 45, 1398105)-net over F4, using
- net defined by OOA [i] based on linear OOA(445, 1398105, F4, 6, 6) (dual of [(1398105, 6), 8388585, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(445, 4194315, F4, 6) (dual of [4194315, 4194270, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(434, 4194304, F4, 5) (dual of [4194304, 4194270, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 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
- OA 3-folding and stacking [i] based on linear OA(445, 4194315, F4, 6) (dual of [4194315, 4194270, 7]-code), using
- net defined by OOA [i] based on linear OOA(445, 1398105, F4, 6, 6) (dual of [(1398105, 6), 8388585, 7]-NRT-code), using
- digital (0, 3, 5)-net over F4, using
(42, 42+6, 4194325)-Net over F4 — Digital
Digital (42, 48, 4194325)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(448, 4194325, F4, 6) (dual of [4194325, 4194277, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(423, 4194304, F4, 3) (dual of [4194304, 4194281, 4]-code or 4194304-cap in PG(22,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, 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
(42, 42+6, large)-Net in Base 4 — Upper bound on s
There is no (42, 48, large)-net in base 4, because
- 4 times m-reduction [i] would yield (42, 44, large)-net in base 4, but