Best Known (50, 50+11, s)-Nets in Base 5
(50, 50+11, 15629)-Net over F5 — Constructive and digital
Digital (50, 61, 15629)-net over F5, using
- net defined by OOA [i] based on linear OOA(561, 15629, F5, 11, 11) (dual of [(15629, 11), 171858, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(561, 78146, F5, 11) (dual of [78146, 78085, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(561, 78150, F5, 11) (dual of [78150, 78089, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(536, 78125, F5, 7) (dual of [78125, 78089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(54, 25, F5, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,5)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(561, 78150, F5, 11) (dual of [78150, 78089, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(561, 78146, F5, 11) (dual of [78146, 78085, 12]-code), using
(50, 50+11, 47363)-Net over F5 — Digital
Digital (50, 61, 47363)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(561, 47363, F5, 11) (dual of [47363, 47302, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(561, 78144, F5, 11) (dual of [78144, 78083, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(557, 78126, F5, 11) (dual of [78126, 78069, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(543, 78126, F5, 7) (dual of [78126, 78083, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(54, 18, F5, 3) (dual of [18, 14, 4]-code or 18-cap in PG(3,5)), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(561, 78144, F5, 11) (dual of [78144, 78083, 12]-code), using
(50, 50+11, large)-Net in Base 5 — Upper bound on s
There is no (50, 61, large)-net in base 5, because
- 9 times m-reduction [i] would yield (50, 52, large)-net in base 5, but