Best Known (61, 61+18, s)-Nets in Base 4
(61, 61+18, 1032)-Net over F4 — Constructive and digital
Digital (61, 79, 1032)-net over F4, using
- 1 times m-reduction [i] based on digital (61, 80, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 20, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 20, 258)-net over F256, using
(61, 61+18, 2051)-Net over F4 — Digital
Digital (61, 79, 2051)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(479, 2051, F4, 2, 18) (dual of [(2051, 2), 4023, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(479, 4102, F4, 18) (dual of [4102, 4023, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 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(473, 4096, F4, 17) (dual of [4096, 4023, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 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
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(479, 4102, F4, 18) (dual of [4102, 4023, 19]-code), using
(61, 61+18, 266297)-Net in Base 4 — Upper bound on s
There is no (61, 79, 266298)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 365377 789391 961190 933311 175236 958162 264212 245101 > 479 [i]