Best Known (94−25, 94, s)-Nets in Base 4
(94−25, 94, 384)-Net over F4 — Constructive and digital
Digital (69, 94, 384)-net over F4, using
- 2 times m-reduction [i] based on digital (69, 96, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 32, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 32, 128)-net over F64, using
(94−25, 94, 450)-Net in Base 4 — Constructive
(69, 94, 450)-net in base 4, using
- 41 times duplication [i] based on (68, 93, 450)-net in base 4, using
- trace code for nets [i] based on (6, 31, 150)-net in base 64, using
- 4 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
- base change [i] based on digital (1, 30, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 30, 150)-net over F128, using
- 4 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
- trace code for nets [i] based on (6, 31, 150)-net in base 64, using
(94−25, 94, 837)-Net over F4 — Digital
Digital (69, 94, 837)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(494, 837, F4, 25) (dual of [837, 743, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(494, 1031, F4, 25) (dual of [1031, 937, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(491, 1025, F4, 25) (dual of [1025, 934, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(481, 1025, F4, 21) (dual of [1025, 944, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(43, 6, F4, 3) (dual of [6, 3, 4]-code or 6-arc in PG(2,4) or 6-cap in PG(2,4)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(494, 1031, F4, 25) (dual of [1031, 937, 26]-code), using
(94−25, 94, 81687)-Net in Base 4 — Upper bound on s
There is no (69, 94, 81688)-net in base 4, because
- 1 times m-reduction [i] would yield (69, 93, 81688)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 98 089669 059633 902759 403899 572690 944053 163236 237958 587374 > 493 [i]