Best Known (196−43, 196, s)-Nets in Base 4
(196−43, 196, 1052)-Net over F4 — Constructive and digital
Digital (153, 196, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 49, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
(196−43, 196, 3896)-Net over F4 — Digital
Digital (153, 196, 3896)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4196, 3896, F4, 43) (dual of [3896, 3700, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4196, 4112, F4, 43) (dual of [4112, 3916, 44]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4194, 4110, F4, 43) (dual of [4110, 3916, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,20]) [i] based on
- linear OA(4193, 4097, F4, 43) (dual of [4097, 3904, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(4181, 4097, F4, 41) (dual of [4097, 3916, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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
- construction X applied to C([0,21]) ⊂ C([0,20]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4194, 4110, F4, 43) (dual of [4110, 3916, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4196, 4112, F4, 43) (dual of [4112, 3916, 44]-code), using
(196−43, 196, 1126994)-Net in Base 4 — Upper bound on s
There is no (153, 196, 1126995)-net in base 4, because
- 1 times m-reduction [i] would yield (153, 195, 1126995)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2521 748565 642267 294176 829578 835588 971308 023955 538551 020007 493807 575669 487363 750271 553771 978568 569465 780841 624147 012704 > 4195 [i]