Best Known (88, 113, s)-Nets in Base 4
(88, 113, 1040)-Net over F4 — Constructive and digital
Digital (88, 113, 1040)-net over F4, using
- 41 times duplication [i] based on digital (87, 112, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 28, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- trace code for nets [i] based on digital (3, 28, 260)-net over F256, using
(88, 113, 2668)-Net over F4 — Digital
Digital (88, 113, 2668)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4113, 2668, F4, 25) (dual of [2668, 2555, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4113, 4113, F4, 25) (dual of [4113, 4000, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(4109, 4097, F4, 25) (dual of [4097, 3988, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(497, 4097, F4, 21) (dual of [4097, 4000, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(44, 16, F4, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,4)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4113, 4113, F4, 25) (dual of [4113, 4000, 26]-code), using
(88, 113, 733603)-Net in Base 4 — Upper bound on s
There is no (88, 113, 733604)-net in base 4, because
- 1 times m-reduction [i] would yield (88, 112, 733604)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 26 960371 235495 445982 380775 666540 618638 676461 337839 836725 273108 294840 > 4112 [i]