Best Known (53, 64, s)-Nets in Base 5
(53, 64, 15646)-Net over F5 — Constructive and digital
Digital (53, 64, 15646)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 21)-net over F5, using
- digital (46, 57, 15625)-net over F5, using
- net defined by OOA [i] based on linear OOA(557, 15625, F5, 11, 11) (dual of [(15625, 11), 171818, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [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]
- OOA 5-folding and stacking with additional row [i] based on linear OA(557, 78126, F5, 11) (dual of [78126, 78069, 12]-code), using
- net defined by OOA [i] based on linear OOA(557, 15625, F5, 11, 11) (dual of [(15625, 11), 171818, 12]-NRT-code), using
(53, 64, 78160)-Net over F5 — Digital
Digital (53, 64, 78160)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(564, 78160, F5, 11) (dual of [78160, 78096, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [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(529, 78125, F5, 6) (dual of [78125, 78096, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(57, 35, F5, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
(53, 64, large)-Net in Base 5 — Upper bound on s
There is no (53, 64, large)-net in base 5, because
- 9 times m-reduction [i] would yield (53, 55, large)-net in base 5, but