Best Known (83−21, 83, s)-Nets in Base 5
(83−21, 83, 313)-Net over F5 — Constructive and digital
Digital (62, 83, 313)-net over F5, using
- 51 times duplication [i] based on digital (61, 82, 313)-net over F5, using
- net defined by OOA [i] based on linear OOA(582, 313, F5, 21, 21) (dual of [(313, 21), 6491, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(582, 3131, F5, 21) (dual of [3131, 3049, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(576, 3125, F5, 19) (dual of [3125, 3049, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(51, 6, F5, 1) (dual of [6, 5, 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(20) ⊂ Ce(18) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(582, 3131, F5, 21) (dual of [3131, 3049, 22]-code), using
- net defined by OOA [i] based on linear OOA(582, 313, F5, 21, 21) (dual of [(313, 21), 6491, 22]-NRT-code), using
(83−21, 83, 2047)-Net over F5 — Digital
Digital (62, 83, 2047)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(583, 2047, F5, 21) (dual of [2047, 1964, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(583, 3133, F5, 21) (dual of [3133, 3050, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(576, 3125, F5, 19) (dual of [3125, 3049, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 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
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(583, 3133, F5, 21) (dual of [3133, 3050, 22]-code), using
(83−21, 83, 610190)-Net in Base 5 — Upper bound on s
There is no (62, 83, 610191)-net in base 5, because
- 1 times m-reduction [i] would yield (62, 82, 610191)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 2067 972116 894233 791881 245653 869969 248181 309381 010377 198553 > 582 [i]