Best Known (254−57, 254, s)-Nets in Base 4
(254−57, 254, 1052)-Net over F4 — Constructive and digital
Digital (197, 254, 1052)-net over F4, using
- 42 times duplication [i] based on digital (195, 252, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 63, 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, 63, 263)-net over F256, using
(254−57, 254, 4103)-Net over F4 — Digital
Digital (197, 254, 4103)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4254, 4103, F4, 57) (dual of [4103, 3849, 58]-code), using
- construction X applied to Ce(56) ⊂ Ce(54) [i] based on
- linear OA(4253, 4096, F4, 57) (dual of [4096, 3843, 58]-code), using an extension Ce(56) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,56], and designed minimum distance d ≥ |I|+1 = 57 [i]
- linear OA(4247, 4096, F4, 55) (dual of [4096, 3849, 56]-code), using an extension Ce(54) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,54], and designed minimum distance d ≥ |I|+1 = 55 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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(56) ⊂ Ce(54) [i] based on
(254−57, 254, 1037328)-Net in Base 4 — Upper bound on s
There is no (197, 254, 1037329)-net in base 4, because
- 1 times m-reduction [i] would yield (197, 253, 1037329)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 209 500235 028689 257079 022182 993383 612330 556210 801483 361775 971370 363195 546839 383096 880178 653186 411334 484238 632920 001088 095298 539184 205059 670934 864670 067456 > 4253 [i]