Best Known (34, 45, s)-Nets in Base 5
(34, 45, 628)-Net over F5 — Constructive and digital
Digital (34, 45, 628)-net over F5, using
- net defined by OOA [i] based on linear OOA(545, 628, F5, 11, 11) (dual of [(628, 11), 6863, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(545, 3141, F5, 11) (dual of [3141, 3096, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(545, 3144, F5, 11) (dual of [3144, 3099, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(541, 3125, F5, 11) (dual of [3125, 3084, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(526, 3125, F5, 7) (dual of [3125, 3099, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-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(545, 3144, F5, 11) (dual of [3144, 3099, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(545, 3141, F5, 11) (dual of [3141, 3096, 12]-code), using
(34, 45, 2704)-Net over F5 — Digital
Digital (34, 45, 2704)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(545, 2704, F5, 11) (dual of [2704, 2659, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(545, 3140, F5, 11) (dual of [3140, 3095, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(541, 3126, F5, 11) (dual of [3126, 3085, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(531, 3126, F5, 7) (dual of [3126, 3095, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(54, 14, F5, 3) (dual of [14, 10, 4]-code or 14-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(545, 3140, F5, 11) (dual of [3140, 3095, 12]-code), using
(34, 45, 921957)-Net in Base 5 — Upper bound on s
There is no (34, 45, 921958)-net in base 5, because
- 1 times m-reduction [i] would yield (34, 44, 921958)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 5 684366 232012 929494 830919 313689 > 544 [i]