Best Known (53−11, 53, s)-Nets in Base 5
(53−11, 53, 3129)-Net over F5 — Constructive and digital
Digital (42, 53, 3129)-net over F5, using
- net defined by OOA [i] based on linear OOA(553, 3129, F5, 11, 11) (dual of [(3129, 11), 34366, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(553, 15646, F5, 11) (dual of [15646, 15593, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(553, 15647, F5, 11) (dual of [15647, 15594, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(531, 15625, F5, 7) (dual of [15625, 15594, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(54, 22, F5, 3) (dual of [22, 18, 4]-code or 22-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(553, 15647, F5, 11) (dual of [15647, 15594, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(553, 15646, F5, 11) (dual of [15646, 15593, 12]-code), using
(53−11, 53, 11323)-Net over F5 — Digital
Digital (42, 53, 11323)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(553, 11323, F5, 11) (dual of [11323, 11270, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(553, 15642, F5, 11) (dual of [15642, 15589, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(537, 15626, F5, 7) (dual of [15626, 15589, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(54, 16, F5, 3) (dual of [16, 12, 4]-code or 16-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(553, 15642, F5, 11) (dual of [15642, 15589, 12]-code), using
(53−11, 53, large)-Net in Base 5 — Upper bound on s
There is no (42, 53, large)-net in base 5, because
- 9 times m-reduction [i] would yield (42, 44, large)-net in base 5, but