Best Known (46, 61, s)-Nets in Base 5
(46, 61, 447)-Net over F5 — Constructive and digital
Digital (46, 61, 447)-net over F5, using
- net defined by OOA [i] based on linear OOA(561, 447, F5, 15, 15) (dual of [(447, 15), 6644, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(561, 3130, F5, 15) (dual of [3130, 3069, 16]-code), using
- 1 times truncation [i] based on linear OA(562, 3131, F5, 16) (dual of [3131, 3069, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(556, 3125, F5, 14) (dual of [3125, 3069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(15) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(562, 3131, F5, 16) (dual of [3131, 3069, 17]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(561, 3130, F5, 15) (dual of [3130, 3069, 16]-code), using
(46, 61, 2376)-Net over F5 — Digital
Digital (46, 61, 2376)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(561, 2376, F5, 15) (dual of [2376, 2315, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(561, 3126, F5, 15) (dual of [3126, 3065, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(561, 3126, F5, 15) (dual of [3126, 3065, 16]-code), using
(46, 61, 827994)-Net in Base 5 — Upper bound on s
There is no (46, 61, 827995)-net in base 5, because
- 1 times m-reduction [i] would yield (46, 60, 827995)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 867362 900848 280558 687821 571934 094867 482517 > 560 [i]