Best Known (165, 212, s)-Nets in Base 4
(165, 212, 1052)-Net over F4 — Constructive and digital
Digital (165, 212, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 53, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
(165, 212, 3873)-Net over F4 — Digital
Digital (165, 212, 3873)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4212, 3873, F4, 47) (dual of [3873, 3661, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(4212, 4109, F4, 47) (dual of [4109, 3897, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(44) [i] based on
- linear OA(4211, 4096, F4, 47) (dual of [4096, 3885, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [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 Ce(46) ⊂ Ce(44) [i] based on
- discarding factors / shortening the dual code based on linear OA(4212, 4109, F4, 47) (dual of [4109, 3897, 48]-code), using
(165, 212, 1048511)-Net in Base 4 — Upper bound on s
There is no (165, 212, 1048512)-net in base 4, because
- 1 times m-reduction [i] would yield (165, 211, 1048512)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 10 830801 040094 876745 675035 381531 584972 111130 238041 051220 676152 541979 350628 946977 552000 861053 170176 332348 752538 677708 534288 679933 > 4211 [i]