Best Known (90−14, 90, s)-Nets in Base 4
(90−14, 90, 9373)-Net over F4 — Constructive and digital
Digital (76, 90, 9373)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (67, 81, 9363)-net over F4, using
- net defined by OOA [i] based on linear OOA(481, 9363, F4, 14, 14) (dual of [(9363, 14), 131001, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(481, 65541, F4, 14) (dual of [65541, 65460, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(481, 65544, F4, 14) (dual of [65544, 65463, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(481, 65544, F4, 14) (dual of [65544, 65463, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(481, 65541, F4, 14) (dual of [65541, 65460, 15]-code), using
- net defined by OOA [i] based on linear OOA(481, 9363, F4, 14, 14) (dual of [(9363, 14), 131001, 15]-NRT-code), using
- digital (2, 9, 10)-net over F4, using
(90−14, 90, 51457)-Net over F4 — Digital
Digital (76, 90, 51457)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(490, 51457, F4, 14) (dual of [51457, 51367, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(490, 65546, F4, 14) (dual of [65546, 65456, 15]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(44, 5, F4, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,4)), using
- dual of repetition code with length 5 [i]
- linear OA(45, 5, F4, 5) (dual of [5, 0, 6]-code or 5-arc in PG(4,4)), using
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(44, 5, F4, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,4)), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(490, 65546, F4, 14) (dual of [65546, 65456, 15]-code), using
(90−14, 90, large)-Net in Base 4 — Upper bound on s
There is no (76, 90, large)-net in base 4, because
- 12 times m-reduction [i] would yield (76, 78, large)-net in base 4, but