Best Known (161, 161+46, s)-Nets in Base 4
(161, 161+46, 1048)-Net over F4 — Constructive and digital
Digital (161, 207, 1048)-net over F4, using
- 1 times m-reduction [i] based on digital (161, 208, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 52, 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, 52, 262)-net over F256, using
(161, 161+46, 3754)-Net over F4 — Digital
Digital (161, 207, 3754)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4207, 3754, F4, 46) (dual of [3754, 3547, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(4207, 4105, F4, 46) (dual of [4105, 3898, 47]-code), using
- construction XX applied to Ce(45) ⊂ Ce(44) ⊂ Ce(42) [i] based on
- 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(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(4193, 4096, F4, 43) (dual of [4096, 3903, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [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(45) ⊂ Ce(44) ⊂ Ce(42) [i] based on
- discarding factors / shortening the dual code based on linear OA(4207, 4105, F4, 46) (dual of [4105, 3898, 47]-code), using
(161, 161+46, 823882)-Net in Base 4 — Upper bound on s
There is no (161, 207, 823883)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 42308 555716 399240 250501 630778 808497 810662 996410 174208 556368 387577 970821 854830 332276 930639 525961 537092 544299 027972 794279 495680 > 4207 [i]