Best Known (69, 69+10, s)-Nets in Base 5
(69, 69+10, 390646)-Net over F5 — Constructive and digital
Digital (69, 79, 390646)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 21)-net over F5, using
- digital (62, 72, 390625)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 390625, F5, 10, 10) (dual of [(390625, 10), 3906178, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(572, 1953125, F5, 10) (dual of [1953125, 1953053, 11]-code), using
- 1 times truncation [i] based on linear OA(573, 1953126, F5, 11) (dual of [1953126, 1953053, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(573, 1953126, F5, 11) (dual of [1953126, 1953053, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(572, 1953125, F5, 10) (dual of [1953125, 1953053, 11]-code), using
- net defined by OOA [i] based on linear OOA(572, 390625, F5, 10, 10) (dual of [(390625, 10), 3906178, 11]-NRT-code), using
(69, 69+10, 1953168)-Net over F5 — Digital
Digital (69, 79, 1953168)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(579, 1953168, F5, 10) (dual of [1953168, 1953089, 11]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(578, 1953166, F5, 10) (dual of [1953166, 1953088, 11]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(573, 1953125, F5, 11) (dual of [1953125, 1953052, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(537, 1953125, F5, 6) (dual of [1953125, 1953088, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(55, 41, F5, 3) (dual of [41, 36, 4]-code or 41-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 Ce(10) ⊂ Ce(5) [i] based on
- linear OA(578, 1953167, F5, 9) (dual of [1953167, 1953089, 10]-code), using Gilbert–Varšamov bound and bm = 578 > Vbs−1(k−1) = 344 244389 682520 195210 911915 692417 997007 056756 661225 [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(578, 1953166, F5, 10) (dual of [1953166, 1953088, 11]-code), using
- construction X with Varšamov bound [i] based on
(69, 69+10, large)-Net in Base 5 — Upper bound on s
There is no (69, 79, large)-net in base 5, because
- 8 times m-reduction [i] would yield (69, 71, large)-net in base 5, but