Best Known (219−49, 219, s)-Nets in Base 4
(219−49, 219, 1048)-Net over F4 — Constructive and digital
Digital (170, 219, 1048)-net over F4, using
- 1 times m-reduction [i] based on digital (170, 220, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 55, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
- trace code for nets [i] based on digital (5, 55, 262)-net over F256, using
(219−49, 219, 3761)-Net over F4 — Digital
Digital (170, 219, 3761)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4219, 3761, F4, 49) (dual of [3761, 3542, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4219, 4105, F4, 49) (dual of [4105, 3886, 50]-code), using
- construction XX applied to Ce(48) ⊂ Ce(46) ⊂ Ce(45) [i] based on
- linear OA(4217, 4096, F4, 49) (dual of [4096, 3879, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- 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(4205, 4096, F4, 46) (dual of [4096, 3891, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(48) ⊂ Ce(46) ⊂ Ce(45) [i] based on
- discarding factors / shortening the dual code based on linear OA(4219, 4105, F4, 49) (dual of [4105, 3886, 50]-code), using
(219−49, 219, 961488)-Net in Base 4 — Upper bound on s
There is no (170, 219, 961489)-net in base 4, because
- 1 times m-reduction [i] would yield (170, 218, 961489)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 177451 432844 230138 753769 753251 807132 189680 896180 381617 938226 772535 419645 582463 758347 468721 070678 663073 645201 800032 817969 887563 618264 > 4218 [i]