Best Known (248−55, 248, s)-Nets in Base 4
(248−55, 248, 1056)-Net over F4 — Constructive and digital
Digital (193, 248, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 62, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
(248−55, 248, 4142)-Net over F4 — Digital
Digital (193, 248, 4142)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4248, 4142, F4, 55) (dual of [4142, 3894, 56]-code), using
- 39 step Varšamov–Edel lengthening with (ri) = (1, 38 times 0) [i] based on linear OA(4247, 4102, F4, 55) (dual of [4102, 3855, 56]-code), using
- construction X applied to Ce(54) ⊂ Ce(53) [i] based on
- 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(4241, 4096, F4, 54) (dual of [4096, 3855, 55]-code), using an extension Ce(53) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,53], and designed minimum distance d ≥ |I|+1 = 54 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(54) ⊂ Ce(53) [i] based on
- 39 step Varšamov–Edel lengthening with (ri) = (1, 38 times 0) [i] based on linear OA(4247, 4102, F4, 55) (dual of [4102, 3855, 56]-code), using
(248−55, 248, 1172226)-Net in Base 4 — Upper bound on s
There is no (193, 248, 1172227)-net in base 4, because
- 1 times m-reduction [i] would yield (193, 247, 1172227)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 51147 875575 378608 115928 866267 148794 702874 348448 798961 357116 051336 801895 930549 597003 995593 560999 212233 013582 677092 527723 008370 755249 822791 952581 920640 > 4247 [i]