Best Known (225−50, 225, s)-Nets in Base 4
(225−50, 225, 1052)-Net over F4 — Constructive and digital
Digital (175, 225, 1052)-net over F4, using
- 41 times duplication [i] based on digital (174, 224, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 56, 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, 56, 263)-net over F256, using
(225−50, 225, 3992)-Net over F4 — Digital
Digital (175, 225, 3992)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4225, 3992, F4, 50) (dual of [3992, 3767, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(4225, 4105, F4, 50) (dual of [4105, 3880, 51]-code), using
- construction XX applied to Ce(49) ⊂ Ce(48) ⊂ Ce(46) [i] based on
- linear OA(4223, 4096, F4, 50) (dual of [4096, 3873, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- 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(40, 7, F4, 0) (dual of [7, 7, 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
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(49) ⊂ Ce(48) ⊂ Ce(46) [i] based on
- discarding factors / shortening the dual code based on linear OA(4225, 4105, F4, 50) (dual of [4105, 3880, 51]-code), using
(225−50, 225, 889271)-Net in Base 4 — Upper bound on s
There is no (175, 225, 889272)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2907 363895 909755 223663 559420 540397 086889 464200 783745 882522 269601 503435 045845 554808 917318 801631 612081 981051 319110 648965 396575 614588 242404 > 4225 [i]