Best Known (67, 67+13, s)-Nets in Base 4
(67, 67+13, 10931)-Net over F4 — Constructive and digital
Digital (67, 80, 10931)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (60, 73, 10922)-net over F4, using
- net defined by OOA [i] based on linear OOA(473, 10922, F4, 13, 13) (dual of [(10922, 13), 141913, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(473, 65533, F4, 13) (dual of [65533, 65460, 14]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(473, 65533, F4, 13) (dual of [65533, 65460, 14]-code), using
- net defined by OOA [i] based on linear OOA(473, 10922, F4, 13, 13) (dual of [(10922, 13), 141913, 14]-NRT-code), using
- digital (1, 7, 9)-net over F4, using
(67, 67+13, 34489)-Net over F4 — Digital
Digital (67, 80, 34489)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(480, 34489, F4, 13) (dual of [34489, 34409, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(480, 65545, F4, 13) (dual of [65545, 65465, 14]-code), using
- (u, u+v)-construction [i] based on
- linear OA(47, 9, F4, 6) (dual of [9, 2, 7]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(48, 10, F4, 7) (dual of [10, 2, 8]-code), using
- 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(47, 9, F4, 6) (dual of [9, 2, 7]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(480, 65545, F4, 13) (dual of [65545, 65465, 14]-code), using
(67, 67+13, large)-Net in Base 4 — Upper bound on s
There is no (67, 80, large)-net in base 4, because
- 11 times m-reduction [i] would yield (67, 69, large)-net in base 4, but