Best Known (32, 32+7, s)-Nets in Base 4
(32, 32+7, 5468)-Net over F4 — Constructive and digital
Digital (32, 39, 5468)-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 (29, 36, 5463)-net over F4, using
- net defined by OOA [i] based on linear OOA(436, 5463, F4, 7, 7) (dual of [(5463, 7), 38205, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(436, 16390, F4, 7) (dual of [16390, 16354, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(436, 16391, F4, 7) (dual of [16391, 16355, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(436, 16391, F4, 7) (dual of [16391, 16355, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(436, 16390, F4, 7) (dual of [16390, 16354, 8]-code), using
- net defined by OOA [i] based on linear OOA(436, 5463, F4, 7, 7) (dual of [(5463, 7), 38205, 8]-NRT-code), using
- digital (0, 3, 5)-net over F4, using
(32, 32+7, 16404)-Net over F4 — Digital
Digital (32, 39, 16404)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(439, 16404, F4, 7) (dual of [16404, 16365, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(437, 16400, F4, 7) (dual of [16400, 16363, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(415, 16, F4, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,4)), using
- dual of repetition code with length 16 [i]
- linear OA(41, 16, F4, 1) (dual of [16, 15, 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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(437, 16402, F4, 6) (dual of [16402, 16365, 7]-code), using Gilbert–Varšamov bound and bm = 437 > Vbs−1(k−1) = 2401 905927 174409 760644 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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(437, 16400, F4, 7) (dual of [16400, 16363, 8]-code), using
- construction X with Varšamov bound [i] based on
(32, 32+7, large)-Net in Base 4 — Upper bound on s
There is no (32, 39, large)-net in base 4, because
- 5 times m-reduction [i] would yield (32, 34, large)-net in base 4, but