Best Known (52, 64, s)-Nets in Base 4
(52, 64, 2731)-Net over F4 — Constructive and digital
Digital (52, 64, 2731)-net over F4, using
- net defined by OOA [i] based on linear OOA(464, 2731, F4, 12, 12) (dual of [(2731, 12), 32708, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(464, 16386, F4, 12) (dual of [16386, 16322, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(464, 16391, F4, 12) (dual of [16391, 16327, 13]-code), using
- 1 times truncation [i] based on linear OA(465, 16392, F4, 13) (dual of [16392, 16327, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(465, 16392, F4, 13) (dual of [16392, 16327, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(464, 16391, F4, 12) (dual of [16391, 16327, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(464, 16386, F4, 12) (dual of [16386, 16322, 13]-code), using
(52, 64, 9365)-Net over F4 — Digital
Digital (52, 64, 9365)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(464, 9365, F4, 12) (dual of [9365, 9301, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(464, 16391, F4, 12) (dual of [16391, 16327, 13]-code), using
- 1 times truncation [i] based on linear OA(465, 16392, F4, 13) (dual of [16392, 16327, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(465, 16392, F4, 13) (dual of [16392, 16327, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(464, 16391, F4, 12) (dual of [16391, 16327, 13]-code), using
(52, 64, 2636776)-Net in Base 4 — Upper bound on s
There is no (52, 64, 2636777)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 340 282715 327393 186859 886825 372878 119308 > 464 [i]