Best Known (14, 14+7, s)-Nets in Base 5
(14, 14+7, 209)-Net over F5 — Constructive and digital
Digital (14, 21, 209)-net over F5, using
- net defined by OOA [i] based on linear OOA(521, 209, F5, 7, 7) (dual of [(209, 7), 1442, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(521, 628, F5, 7) (dual of [628, 607, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(521, 629, F5, 7) (dual of [629, 608, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(521, 625, F5, 7) (dual of [625, 604, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(517, 625, F5, 6) (dual of [625, 608, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 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
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(521, 629, F5, 7) (dual of [629, 608, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(521, 628, F5, 7) (dual of [628, 607, 8]-code), using
(14, 14+7, 404)-Net over F5 — Digital
Digital (14, 21, 404)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(521, 404, F5, 7) (dual of [404, 383, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(521, 624, F5, 7) (dual of [624, 603, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(521, 624, F5, 7) (dual of [624, 603, 8]-code), using
(14, 14+7, 20753)-Net in Base 5 — Upper bound on s
There is no (14, 21, 20754)-net in base 5, because
- 1 times m-reduction [i] would yield (14, 20, 20754)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 95 377077 944809 > 520 [i]