Best Known (149−29, 149, s)-Nets in Base 5
(149−29, 149, 1118)-Net over F5 — Constructive and digital
Digital (120, 149, 1118)-net over F5, using
- 54 times duplication [i] based on digital (116, 145, 1118)-net over F5, using
- net defined by OOA [i] based on linear OOA(5145, 1118, F5, 29, 29) (dual of [(1118, 29), 32277, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(5145, 15653, F5, 29) (dual of [15653, 15508, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5145, 15655, F5, 29) (dual of [15655, 15510, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- linear OA(5139, 15625, F5, 29) (dual of [15625, 15486, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(5115, 15625, F5, 24) (dual of [15625, 15510, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(5145, 15655, F5, 29) (dual of [15655, 15510, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(5145, 15653, F5, 29) (dual of [15653, 15508, 30]-code), using
- net defined by OOA [i] based on linear OOA(5145, 1118, F5, 29, 29) (dual of [(1118, 29), 32277, 30]-NRT-code), using
(149−29, 149, 15666)-Net over F5 — Digital
Digital (120, 149, 15666)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5149, 15666, F5, 29) (dual of [15666, 15517, 30]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5148, 15664, F5, 29) (dual of [15664, 15516, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(5139, 15625, F5, 29) (dual of [15625, 15486, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(59, 39, F5, 5) (dual of [39, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(59, 44, F5, 5) (dual of [44, 35, 6]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(5148, 15665, F5, 28) (dual of [15665, 15517, 29]-code), using Gilbert–Varšamov bound and bm = 5148 > Vbs−1(k−1) = 296114 373341 506585 383410 706326 403054 241556 154364 513015 044560 890026 227565 733661 107911 608737 854238 624577 [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(5148, 15664, F5, 29) (dual of [15664, 15516, 30]-code), using
- construction X with Varšamov bound [i] based on
(149−29, 149, large)-Net in Base 5 — Upper bound on s
There is no (120, 149, large)-net in base 5, because
- 27 times m-reduction [i] would yield (120, 122, large)-net in base 5, but