Best Known (83, 83+21, s)-Nets in Base 5
(83, 83+21, 1565)-Net over F5 — Constructive and digital
Digital (83, 104, 1565)-net over F5, using
- 51 times duplication [i] based on digital (82, 103, 1565)-net over F5, using
- net defined by OOA [i] based on linear OOA(5103, 1565, F5, 21, 21) (dual of [(1565, 21), 32762, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5103, 15651, F5, 21) (dual of [15651, 15548, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(54, 24, F5, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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 XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(5103, 15651, F5, 21) (dual of [15651, 15548, 22]-code), using
- net defined by OOA [i] based on linear OOA(5103, 1565, F5, 21, 21) (dual of [(1565, 21), 32762, 22]-NRT-code), using
(83, 83+21, 12185)-Net over F5 — Digital
Digital (83, 104, 12185)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5104, 12185, F5, 21) (dual of [12185, 12081, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5104, 15647, F5, 21) (dual of [15647, 15543, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(597, 15626, F5, 21) (dual of [15626, 15529, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(573, 15626, F5, 15) (dual of [15626, 15553, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(57, 21, F5, 5) (dual of [21, 14, 6]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5104, 15647, F5, 21) (dual of [15647, 15543, 22]-code), using
(83, 83+21, large)-Net in Base 5 — Upper bound on s
There is no (83, 104, large)-net in base 5, because
- 19 times m-reduction [i] would yield (83, 85, large)-net in base 5, but