Best Known (82, 82+15, s)-Nets in Base 4
(82, 82+15, 9372)-Net over F4 — Constructive and digital
Digital (82, 97, 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 (74, 89, 9363)-net over F4, using
- net defined by OOA [i] based on linear OOA(489, 9363, F4, 15, 15) (dual of [(9363, 15), 140356, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(489, 65542, F4, 15) (dual of [65542, 65453, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 65544, F4, 15) (dual of [65544, 65455, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(489, 65544, F4, 15) (dual of [65544, 65455, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(489, 65542, F4, 15) (dual of [65542, 65453, 16]-code), using
- net defined by OOA [i] based on linear OOA(489, 9363, F4, 15, 15) (dual of [(9363, 15), 140356, 16]-NRT-code), using
- digital (1, 8, 9)-net over F4, using
(82, 82+15, 52746)-Net over F4 — Digital
Digital (82, 97, 52746)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(497, 52746, F4, 15) (dual of [52746, 52649, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(497, 65546, F4, 15) (dual of [65546, 65449, 16]-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(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(497, 65546, F4, 15) (dual of [65546, 65449, 16]-code), using
(82, 82+15, large)-Net in Base 4 — Upper bound on s
There is no (82, 97, large)-net in base 4, because
- 13 times m-reduction [i] would yield (82, 84, large)-net in base 4, but