Best Known (74−15, 74, s)-Nets in Base 5
(74−15, 74, 2234)-Net over F5 — Constructive and digital
Digital (59, 74, 2234)-net over F5, using
- net defined by OOA [i] based on linear OOA(574, 2234, F5, 15, 15) (dual of [(2234, 15), 33436, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(574, 15639, F5, 15) (dual of [15639, 15565, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(573, 15626, F5, 15) (dual of [15626, 15553, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(561, 15626, F5, 13) (dual of [15626, 15565, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(574, 15639, F5, 15) (dual of [15639, 15565, 16]-code), using
(74−15, 74, 11913)-Net over F5 — Digital
Digital (59, 74, 11913)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(574, 11913, F5, 15) (dual of [11913, 11839, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(574, 15639, F5, 15) (dual of [15639, 15565, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(573, 15626, F5, 15) (dual of [15626, 15553, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(561, 15626, F5, 13) (dual of [15626, 15565, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(574, 15639, F5, 15) (dual of [15639, 15565, 16]-code), using
(74−15, 74, large)-Net in Base 5 — Upper bound on s
There is no (59, 74, large)-net in base 5, because
- 13 times m-reduction [i] would yield (59, 61, large)-net in base 5, but