Best Known (92, 92+19, s)-Nets in Base 4
(92, 92+19, 1835)-Net over F4 — Constructive and digital
Digital (92, 111, 1835)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (80, 99, 1821)-net over F4, using
- net defined by OOA [i] based on linear OOA(499, 1821, F4, 19, 19) (dual of [(1821, 19), 34500, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(499, 16390, F4, 19) (dual of [16390, 16291, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(499, 16391, F4, 19) (dual of [16391, 16292, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(499, 16391, F4, 19) (dual of [16391, 16292, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(499, 16390, F4, 19) (dual of [16390, 16291, 20]-code), using
- net defined by OOA [i] based on linear OOA(499, 1821, F4, 19, 19) (dual of [(1821, 19), 34500, 20]-NRT-code), using
- digital (3, 12, 14)-net over F4, using
(92, 92+19, 16433)-Net over F4 — Digital
Digital (92, 111, 16433)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4111, 16433, F4, 19) (dual of [16433, 16322, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4109, 16429, F4, 19) (dual of [16429, 16320, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(410, 45, F4, 5) (dual of [45, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(4109, 16431, F4, 18) (dual of [16431, 16322, 19]-code), using Gilbert–Varšamov bound and bm = 4109 > Vbs−1(k−1) = 166872 685441 752513 722190 940821 059033 293816 304441 635352 717448 085054 [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(4109, 16429, F4, 19) (dual of [16429, 16320, 20]-code), using
- construction X with Varšamov bound [i] based on
(92, 92+19, large)-Net in Base 4 — Upper bound on s
There is no (92, 111, large)-net in base 4, because
- 17 times m-reduction [i] would yield (92, 94, large)-net in base 4, but