Best Known (158, 158+43, s)-Nets in Base 4
(158, 158+43, 1056)-Net over F4 — Constructive and digital
Digital (158, 201, 1056)-net over F4, using
- 41 times duplication [i] based on digital (157, 200, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 50, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
- trace code for nets [i] based on digital (7, 50, 264)-net over F256, using
(158, 158+43, 4277)-Net over F4 — Digital
Digital (158, 201, 4277)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4201, 4277, F4, 43) (dual of [4277, 4076, 44]-code), using
- 160 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 43 times 0, 1, 70 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
- 160 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 43 times 0, 1, 70 times 0) [i] based on linear OA(4194, 4110, F4, 43) (dual of [4110, 3916, 44]-code), using
(158, 158+43, 1567730)-Net in Base 4 — Upper bound on s
There is no (158, 201, 1567731)-net in base 4, because
- 1 times m-reduction [i] would yield (158, 200, 1567731)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 582282 789404 601477 742901 976891 525587 453161 336994 927332 407580 330190 106952 205721 946198 920077 535603 506623 634473 519603 398896 > 4200 [i]