Best Known (45−15, 45, s)-Nets in Base 25
(45−15, 45, 2233)-Net over F25 — Constructive and digital
Digital (30, 45, 2233)-net over F25, using
- 251 times duplication [i] based on digital (29, 44, 2233)-net over F25, using
- net defined by OOA [i] based on linear OOA(2544, 2233, F25, 15, 15) (dual of [(2233, 15), 33451, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2544, 15632, F25, 15) (dual of [15632, 15588, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2544, 15633, F25, 15) (dual of [15633, 15589, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(2543, 15626, F25, 15) (dual of [15626, 15583, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2537, 15626, F25, 13) (dual of [15626, 15589, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(251, 7, F25, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 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(2544, 15633, F25, 15) (dual of [15633, 15589, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2544, 15632, F25, 15) (dual of [15632, 15588, 16]-code), using
- net defined by OOA [i] based on linear OOA(2544, 2233, F25, 15, 15) (dual of [(2233, 15), 33451, 16]-NRT-code), using
(45−15, 45, 12720)-Net over F25 — Digital
Digital (30, 45, 12720)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2545, 12720, F25, 15) (dual of [12720, 12675, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2545, 15636, F25, 15) (dual of [15636, 15591, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(2543, 15625, F25, 15) (dual of [15625, 15582, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2534, 15625, F25, 12) (dual of [15625, 15591, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(252, 11, F25, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(2545, 15636, F25, 15) (dual of [15636, 15591, 16]-code), using
(45−15, 45, large)-Net in Base 25 — Upper bound on s
There is no (30, 45, large)-net in base 25, because
- 13 times m-reduction [i] would yield (30, 32, large)-net in base 25, but