Best Known (156, 156+43, s)-Nets in Base 4
(156, 156+43, 1052)-Net over F4 — Constructive and digital
Digital (156, 199, 1052)-net over F4, using
- 1 times m-reduction [i] based on digital (156, 200, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 50, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
- trace code for nets [i] based on digital (6, 50, 263)-net over F256, using
(156, 156+43, 4160)-Net over F4 — Digital
Digital (156, 199, 4160)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4199, 4160, F4, 43) (dual of [4160, 3961, 44]-code), using
- 45 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0) [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
- 45 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0) [i] based on linear OA(4194, 4110, F4, 43) (dual of [4110, 3916, 44]-code), using
(156, 156+43, 1373825)-Net in Base 4 — Upper bound on s
There is no (156, 199, 1373826)-net in base 4, because
- 1 times m-reduction [i] would yield (156, 198, 1373826)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 161391 867711 235640 725332 107189 769989 947863 559811 681407 875980 760276 109640 567306 795747 072501 089915 003764 433283 951631 304864 > 4198 [i]