Best Known (49, 59, s)-Nets in Base 5
(49, 59, 15628)-Net over F5 — Constructive and digital
Digital (49, 59, 15628)-net over F5, using
- 1 times m-reduction [i] based on digital (49, 60, 15628)-net over F5, using
- net defined by OOA [i] based on linear OOA(560, 15628, F5, 11, 11) (dual of [(15628, 11), 171848, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(560, 78141, F5, 11) (dual of [78141, 78081, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(560, 78142, F5, 11) (dual of [78142, 78082, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(53, 17, F5, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(560, 78142, F5, 11) (dual of [78142, 78082, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(560, 78141, F5, 11) (dual of [78141, 78081, 12]-code), using
- net defined by OOA [i] based on linear OOA(560, 15628, F5, 11, 11) (dual of [(15628, 11), 171848, 12]-NRT-code), using
(49, 59, 78143)-Net over F5 — Digital
Digital (49, 59, 78143)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(559, 78143, F5, 10) (dual of [78143, 78084, 11]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(558, 78141, F5, 10) (dual of [78141, 78083, 11]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(515, 16, F5, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,5)), using
- dual of repetition code with length 16 [i]
- linear OA(51, 16, F5, 1) (dual of [16, 15, 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 X4 applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(558, 78142, F5, 9) (dual of [78142, 78084, 10]-code), using Gilbert–Varšamov bound and bm = 558 > Vbs−1(k−1) = 2258 637933 317132 278470 620900 305806 471125 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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
- linear OA(558, 78141, F5, 10) (dual of [78141, 78083, 11]-code), using
- construction X with Varšamov bound [i] based on
(49, 59, large)-Net in Base 5 — Upper bound on s
There is no (49, 59, large)-net in base 5, because
- 8 times m-reduction [i] would yield (49, 51, large)-net in base 5, but