Best Known (63, 63+20, s)-Nets in Base 5
(63, 63+20, 313)-Net over F5 — Constructive and digital
Digital (63, 83, 313)-net over F5, using
- 51 times duplication [i] based on digital (62, 82, 313)-net over F5, using
- t-expansion [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
- t-expansion [i] based on digital (61, 82, 313)-net over F5, using
(63, 63+20, 2873)-Net over F5 — Digital
Digital (63, 83, 2873)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(583, 2873, F5, 20) (dual of [2873, 2790, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(583, 3132, F5, 20) (dual of [3132, 3049, 21]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(581, 3126, F5, 21) (dual of [3126, 3045, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(571, 3126, F5, 17) (dual of [3126, 3055, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(52, 6, F5, 2) (dual of [6, 4, 3]-code or 6-arc in PG(1,5)), using
- extended Reed–Solomon code RSe(4,5) [i]
- Hamming code H(2,5) [i]
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(583, 3132, F5, 20) (dual of [3132, 3049, 21]-code), using
(63, 63+20, 716742)-Net in Base 5 — Upper bound on s
There is no (63, 83, 716743)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 10339 841855 920549 770586 483124 312870 611886 377076 473651 059673 > 583 [i]