Best Known (93−21, 93, s)-Nets in Base 4
(93−21, 93, 1036)-Net over F4 — Constructive and digital
Digital (72, 93, 1036)-net over F4, using
- 41 times duplication [i] based on digital (71, 92, 1036)-net over F4, using
- trace code for nets [i] based on digital (2, 23, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- trace code for nets [i] based on digital (2, 23, 259)-net over F256, using
(93−21, 93, 2160)-Net over F4 — Digital
Digital (72, 93, 2160)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(493, 2160, F4, 21) (dual of [2160, 2067, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(493, 4105, F4, 21) (dual of [4105, 4012, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- linear OA(491, 4096, F4, 21) (dual of [4096, 4005, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(485, 4096, F4, 19) (dual of [4096, 4011, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(479, 4096, F4, 18) (dual of [4096, 4017, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(493, 4105, F4, 21) (dual of [4105, 4012, 22]-code), using
(93−21, 93, 522156)-Net in Base 4 — Upper bound on s
There is no (72, 93, 522157)-net in base 4, because
- 1 times m-reduction [i] would yield (72, 92, 522157)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 24 520296 126329 131848 973554 705306 730719 939493 556899 189450 > 492 [i]