Best Known (113, 136, s)-Nets in Base 5
(113, 136, 7106)-Net over F5 — Constructive and digital
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
(113, 136, 67564)-Net over F5 — Digital
Digital (113, 136, 67564)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5136, 67564, F5, 23) (dual of [67564, 67428, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5136, 78153, F5, 23) (dual of [78153, 78017, 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(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([0,11]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5136, 78153, F5, 23) (dual of [78153, 78017, 24]-code), using
(113, 136, large)-Net in Base 5 — Upper bound on s
There is no (113, 136, large)-net in base 5, because
- 21 times m-reduction [i] would yield (113, 115, large)-net in base 5, but