Best Known (90, 90+19, s)-Nets in Base 4
(90, 90+19, 1830)-Net over F4 — Constructive and digital
Digital (90, 109, 1830)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-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 (1, 10, 9)-net over F4, using
(90, 90+19, 15971)-Net over F4 — Digital
Digital (90, 109, 15971)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4109, 15971, F4, 19) (dual of [15971, 15862, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4109, 16423, F4, 19) (dual of [16423, 16314, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(499, 16385, F4, 19) (dual of [16385, 16286, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(471, 16385, F4, 13) (dual of [16385, 16314, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(410, 38, F4, 5) (dual of [38, 28, 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 C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4109, 16423, F4, 19) (dual of [16423, 16314, 20]-code), using
(90, 90+19, large)-Net in Base 4 — Upper bound on s
There is no (90, 109, large)-net in base 4, because
- 17 times m-reduction [i] would yield (90, 92, large)-net in base 4, but