Best Known (117−26, 117, s)-Nets in Base 4
(117−26, 117, 1040)-Net over F4 — Constructive and digital
Digital (91, 117, 1040)-net over F4, using
- 41 times duplication [i] based on digital (90, 116, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 29, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- trace code for nets [i] based on digital (3, 29, 260)-net over F256, using
(117−26, 117, 2637)-Net over F4 — Digital
Digital (91, 117, 2637)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4117, 2637, F4, 26) (dual of [2637, 2520, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4117, 4105, F4, 26) (dual of [4105, 3988, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4115, 4096, F4, 26) (dual of [4096, 3981, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4109, 4096, F4, 25) (dual of [4096, 3987, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4103, 4096, F4, 23) (dual of [4096, 3993, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(25) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4117, 4105, F4, 26) (dual of [4105, 3988, 27]-code), using
(117−26, 117, 495240)-Net in Base 4 — Upper bound on s
There is no (91, 117, 495241)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 27607 653062 677722 148549 036356 635429 242146 603696 366805 253720 361127 694080 > 4117 [i]