Best Known (80, 80+19, s)-Nets in Base 5
(80, 80+19, 1739)-Net over F5 — Constructive and digital
Digital (80, 99, 1739)-net over F5, using
- 52 times duplication [i] based on digital (78, 97, 1739)-net over F5, using
- net defined by OOA [i] based on linear OOA(597, 1739, F5, 19, 19) (dual of [(1739, 19), 32944, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(597, 15652, F5, 19) (dual of [15652, 15555, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(597, 15655, F5, 19) (dual of [15655, 15558, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(591, 15625, F5, 19) (dual of [15625, 15534, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(597, 15655, F5, 19) (dual of [15655, 15558, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(597, 15652, F5, 19) (dual of [15652, 15555, 20]-code), using
- net defined by OOA [i] based on linear OOA(597, 1739, F5, 19, 19) (dual of [(1739, 19), 32944, 20]-NRT-code), using
(80, 80+19, 15659)-Net over F5 — Digital
Digital (80, 99, 15659)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(599, 15659, F5, 19) (dual of [15659, 15560, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(597, 15655, F5, 19) (dual of [15655, 15558, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(591, 15625, F5, 19) (dual of [15625, 15534, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(597, 15657, F5, 18) (dual of [15657, 15560, 19]-code), using Gilbert–Varšamov bound and bm = 597 > Vbs−1(k−1) = 9 769430 352196 381753 561125 904092 064489 021480 151655 493943 719775 568225 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(597, 15655, F5, 19) (dual of [15655, 15558, 20]-code), using
- construction X with Varšamov bound [i] based on
(80, 80+19, large)-Net in Base 5 — Upper bound on s
There is no (80, 99, large)-net in base 5, because
- 17 times m-reduction [i] would yield (80, 82, large)-net in base 5, but