Best Known (104, 104+15, s)-Nets in Base 5
(104, 104+15, 279044)-Net over F5 — Constructive and digital
Digital (104, 119, 279044)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 25)-net over F5, using
- digital (94, 109, 279019)-net over F5, using
- net defined by OOA [i] based on linear OOA(5109, 279019, F5, 15, 15) (dual of [(279019, 15), 4185176, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5109, 1953134, F5, 15) (dual of [1953134, 1953025, 16]-code), using
- 1 times truncation [i] based on linear OA(5110, 1953135, F5, 16) (dual of [1953135, 1953025, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 10, F5, 1) (dual of [10, 9, 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 Ce(15) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(5110, 1953135, F5, 16) (dual of [1953135, 1953025, 17]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5109, 1953134, F5, 15) (dual of [1953134, 1953025, 16]-code), using
- net defined by OOA [i] based on linear OOA(5109, 279019, F5, 15, 15) (dual of [(279019, 15), 4185176, 16]-NRT-code), using
(104, 104+15, 1953181)-Net over F5 — Digital
Digital (104, 119, 1953181)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5119, 1953181, F5, 15) (dual of [1953181, 1953062, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5118, 1953179, F5, 15) (dual of [1953179, 1953061, 16]-code), using
- construction X applied to Ce(15) ⊂ Ce(8) [i] based on
- linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(564, 1953125, F5, 9) (dual of [1953125, 1953061, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(59, 54, F5, 5) (dual of [54, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- construction X applied to Ce(15) ⊂ Ce(8) [i] based on
- linear OA(5118, 1953180, F5, 14) (dual of [1953180, 1953062, 15]-code), using Gilbert–Varšamov bound and bm = 5118 > Vbs−1(k−1) = 64 882795 148864 698538 955477 829297 899618 195703 811833 516535 981960 079650 372966 459805 [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(5118, 1953179, F5, 15) (dual of [1953179, 1953061, 16]-code), using
- construction X with Varšamov bound [i] based on
(104, 104+15, large)-Net in Base 5 — Upper bound on s
There is no (104, 119, large)-net in base 5, because
- 13 times m-reduction [i] would yield (104, 106, large)-net in base 5, but