Best Known (81, 81+18, s)-Nets in Base 4
(81, 81+18, 1824)-Net over F4 — Constructive and digital
Digital (81, 99, 1824)-net over F4, using
- net defined by OOA [i] based on linear OOA(499, 1824, F4, 18, 18) (dual of [(1824, 18), 32733, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(499, 16416, F4, 18) (dual of [16416, 16317, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(499, 16419, F4, 18) (dual of [16419, 16320, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(499, 16419, F4, 18) (dual of [16419, 16320, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(499, 16416, F4, 18) (dual of [16416, 16317, 19]-code), using
(81, 81+18, 11029)-Net over F4 — Digital
Digital (81, 99, 11029)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(499, 11029, F4, 18) (dual of [11029, 10930, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(499, 16419, F4, 18) (dual of [16419, 16320, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(499, 16419, F4, 18) (dual of [16419, 16320, 19]-code), using
(81, 81+18, 5798151)-Net in Base 4 — Upper bound on s
There is no (81, 99, 5798152)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 401734 699514 949625 778643 313016 567241 077052 370048 829227 511776 > 499 [i]