Best Known (149−25, 149, s)-Nets in Base 5
(149−25, 149, 6513)-Net over F5 — Constructive and digital
Digital (124, 149, 6513)-net over F5, using
- 53 times duplication [i] based on digital (121, 146, 6513)-net over F5, using
- net defined by OOA [i] based on linear OOA(5146, 6513, F5, 25, 25) (dual of [(6513, 25), 162679, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5146, 78157, F5, 25) (dual of [78157, 78011, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5146, 78159, F5, 25) (dual of [78159, 78013, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(5141, 78126, F5, 25) (dual of [78126, 77985, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(5113, 78126, F5, 21) (dual of [78126, 78013, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(55, 33, F5, 3) (dual of [33, 28, 4]-code or 33-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5146, 78159, F5, 25) (dual of [78159, 78013, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(5146, 78157, F5, 25) (dual of [78157, 78011, 26]-code), using
- net defined by OOA [i] based on linear OOA(5146, 6513, F5, 25, 25) (dual of [(6513, 25), 162679, 26]-NRT-code), using
(149−25, 149, 74135)-Net over F5 — Digital
Digital (124, 149, 74135)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5149, 74135, F5, 25) (dual of [74135, 73986, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5149, 78162, F5, 25) (dual of [78162, 78013, 26]-code), using
- 3 times code embedding in larger space [i] based on linear OA(5146, 78159, F5, 25) (dual of [78159, 78013, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(5141, 78126, F5, 25) (dual of [78126, 77985, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(5113, 78126, F5, 21) (dual of [78126, 78013, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(55, 33, F5, 3) (dual of [33, 28, 4]-code or 33-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(5146, 78159, F5, 25) (dual of [78159, 78013, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(5149, 78162, F5, 25) (dual of [78162, 78013, 26]-code), using
(149−25, 149, large)-Net in Base 5 — Upper bound on s
There is no (124, 149, large)-net in base 5, because
- 23 times m-reduction [i] would yield (124, 126, large)-net in base 5, but