Best Known (220−49, 220, s)-Nets in Base 4
(220−49, 220, 1052)-Net over F4 — Constructive and digital
Digital (171, 220, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 55, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
(220−49, 220, 3874)-Net over F4 — Digital
Digital (171, 220, 3874)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4220, 3874, F4, 49) (dual of [3874, 3654, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4220, 4103, F4, 49) (dual of [4103, 3883, 50]-code), using
- construction X applied to C([0,24]) ⊂ C([0,22]) [i] based on
- linear OA(4217, 4097, F4, 49) (dual of [4097, 3880, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- linear OA(4205, 4097, F4, 45) (dual of [4097, 3892, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (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([0,24]) ⊂ C([0,22]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4220, 4103, F4, 49) (dual of [4103, 3883, 50]-code), using
(220−49, 220, 1018663)-Net in Base 4 — Upper bound on s
There is no (171, 220, 1018664)-net in base 4, because
- 1 times m-reduction [i] would yield (171, 219, 1018664)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 709817 928411 104576 531514 701379 493450 651535 508946 390330 632693 324782 401354 473192 953113 240746 426551 047967 292791 294092 533907 684355 651764 > 4219 [i]