Best Known (234−33, 234, s)-Nets in Base 4
(234−33, 234, 16393)-Net over F4 — Constructive and digital
Digital (201, 234, 16393)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 17, 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 (184, 217, 16384)-net over F4, using
- net defined by OOA [i] based on linear OOA(4217, 16384, F4, 33, 33) (dual of [(16384, 33), 540455, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4217, 262145, F4, 33) (dual of [262145, 261928, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(4217, 262145, F4, 33) (dual of [262145, 261928, 34]-code), using
- net defined by OOA [i] based on linear OOA(4217, 16384, F4, 33, 33) (dual of [(16384, 33), 540455, 34]-NRT-code), using
- digital (1, 17, 9)-net over F4, using
(234−33, 234, 138674)-Net over F4 — Digital
Digital (201, 234, 138674)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4234, 138674, F4, 33) (dual of [138674, 138440, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4234, 262213, F4, 33) (dual of [262213, 261979, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4217, 262145, F4, 33) (dual of [262145, 261928, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(417, 68, F4, 7) (dual of [68, 51, 8]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(43, 17, F4, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- linear OA(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- linear OA(49, 17, F4, 7) (dual of [17, 8, 8]-code), using
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- generalized (u, u+v)-construction [i] based on
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4234, 262213, F4, 33) (dual of [262213, 261979, 34]-code), using
(234−33, 234, large)-Net in Base 4 — Upper bound on s
There is no (201, 234, large)-net in base 4, because
- 31 times m-reduction [i] would yield (201, 203, large)-net in base 4, but