Best Known (137−23, 137, s)-Nets in Base 5
(137−23, 137, 7106)-Net over F5 — Constructive and digital
Digital (114, 137, 7106)-net over F5, using
- 51 times duplication [i] based on digital (113, 136, 7106)-net over F5, using
- net defined by OOA [i] based on linear OOA(5136, 7106, F5, 23, 23) (dual of [(7106, 23), 163302, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5136, 78167, F5, 23) (dual of [78167, 78031, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5136, 78169, F5, 23) (dual of [78169, 78033, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(5127, 78125, F5, 23) (dual of [78125, 77998, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(59, 44, F5, 5) (dual of [44, 35, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5136, 78169, F5, 23) (dual of [78169, 78033, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5136, 78167, F5, 23) (dual of [78167, 78031, 24]-code), using
- net defined by OOA [i] based on linear OOA(5136, 7106, F5, 23, 23) (dual of [(7106, 23), 163302, 24]-NRT-code), using
(137−23, 137, 72947)-Net over F5 — Digital
Digital (114, 137, 72947)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5137, 72947, F5, 23) (dual of [72947, 72810, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5137, 78164, F5, 23) (dual of [78164, 78027, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(5127, 78126, F5, 23) (dual of [78126, 77999, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(599, 78126, F5, 17) (dual of [78126, 78027, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(510, 38, F5, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(510, 52, F5, 5) (dual of [52, 42, 6]-code), using
- trace code [i] based on linear OA(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using
- extended Reed–Solomon code RSe(21,25) [i]
- the expurgated narrow-sense BCH-code C(I) with length 26 | 252−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- algebraic-geometric code AG(F,10P) with degPÂ =Â 2 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using the rational function field F25(x) [i]
- algebraic-geometric code AG(F, Q+6P) with degQ = 2 and degPÂ =Â 3 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26 (see above)
- trace code [i] based on linear OA(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using
- discarding factors / shortening the dual code based on linear OA(510, 52, F5, 5) (dual of [52, 42, 6]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5137, 78164, F5, 23) (dual of [78164, 78027, 24]-code), using
(137−23, 137, large)-Net in Base 5 — Upper bound on s
There is no (114, 137, large)-net in base 5, because
- 21 times m-reduction [i] would yield (114, 116, large)-net in base 5, but