Best Known (75, 75+14, s)-Nets in Base 4
(75, 75+14, 9372)-Net over F4 — Constructive and digital
Digital (75, 89, 9372)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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 (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 (1, 8, 9)-net over F4, using
(75, 75+14, 45842)-Net over F4 — Digital
Digital (75, 89, 45842)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(489, 45842, F4, 14) (dual of [45842, 45753, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 65546, F4, 14) (dual of [65546, 65457, 15]-code), using
- (u, u+v)-construction [i] based on
- linear OA(48, 10, F4, 7) (dual of [10, 2, 8]-code), using
- repeating each code word 2 times [i] based on linear OA(43, 5, F4, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), using
- extended Reed–Solomon code RSe(2,4) [i]
- Simplex code S(2,4) [i]
- repeating each code word 2 times [i] based on linear OA(43, 5, F4, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), 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(48, 10, F4, 7) (dual of [10, 2, 8]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(489, 65546, F4, 14) (dual of [65546, 65457, 15]-code), using
(75, 75+14, large)-Net in Base 4 — Upper bound on s
There is no (75, 89, large)-net in base 4, because
- 12 times m-reduction [i] would yield (75, 77, large)-net in base 4, but