Best Known (98−19, 98, s)-Nets in Base 5
(98−19, 98, 1739)-Net over F5 — Constructive and digital
Digital (79, 98, 1739)-net over F5, using
- 51 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
(98−19, 98, 15657)-Net over F5 — Digital
Digital (79, 98, 15657)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(598, 15657, F5, 19) (dual of [15657, 15559, 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, 15656, F5, 18) (dual of [15656, 15559, 19]-code), using Gilbert–Varšamov bound and bm = 597 > Vbs−1(k−1) = 9 758822 428038 598157 220133 455138 516909 720110 032514 725888 877394 683565 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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
(98−19, 98, large)-Net in Base 5 — Upper bound on s
There is no (79, 98, large)-net in base 5, because
- 17 times m-reduction [i] would yield (79, 81, large)-net in base 5, but