Best Known (138−21, 138, s)-Nets in Base 5
(138−21, 138, 39066)-Net over F5 — Constructive and digital
Digital (117, 138, 39066)-net over F5, using
- 52 times duplication [i] based on digital (115, 136, 39066)-net over F5, using
- net defined by OOA [i] based on linear OOA(5136, 39066, F5, 21, 21) (dual of [(39066, 21), 820250, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5136, 390661, F5, 21) (dual of [390661, 390525, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5136, 390664, F5, 21) (dual of [390664, 390528, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(57, 39, F5, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5136, 390664, F5, 21) (dual of [390664, 390528, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5136, 390661, F5, 21) (dual of [390661, 390525, 22]-code), using
- net defined by OOA [i] based on linear OOA(5136, 39066, F5, 21, 21) (dual of [(39066, 21), 820250, 22]-NRT-code), using
(138−21, 138, 217305)-Net over F5 — Digital
Digital (117, 138, 217305)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5138, 217305, F5, 21) (dual of [217305, 217167, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5138, 390653, F5, 21) (dual of [390653, 390515, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(5129, 390626, F5, 21) (dual of [390626, 390497, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(597, 390626, F5, 15) (dual of [390626, 390529, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(59, 27, F5, 5) (dual of [27, 18, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([1,2]) [i] based on
- linear OA(59, 26, F5, 5) (dual of [26, 17, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 26 | 54−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(58, 26, F5, 4) (dual of [26, 18, 5]-code), using the narrow-sense BCH-code C(I) with length 26 | 54−1, defining interval I = [1,2], and minimum distance d ≥ |{5,16,1}| + |{−9,−6,−3,0}∖{−3,−9}| = 5 (general Roos-bound) [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
- construction X applied to C([0,2]) ⊂ C([1,2]) [i] based on
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5138, 390653, F5, 21) (dual of [390653, 390515, 22]-code), using
(138−21, 138, large)-Net in Base 5 — Upper bound on s
There is no (117, 138, large)-net in base 5, because
- 19 times m-reduction [i] would yield (117, 119, large)-net in base 5, but