Best Known (50, 65, s)-Nets in Base 5
(50, 65, 449)-Net over F5 — Constructive and digital
Digital (50, 65, 449)-net over F5, using
- net defined by OOA [i] based on linear OOA(565, 449, F5, 15, 15) (dual of [(449, 15), 6670, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(565, 3144, F5, 15) (dual of [3144, 3079, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(565, 3150, F5, 15) (dual of [3150, 3085, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] 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]
- linear OA(541, 3126, F5, 11) (dual of [3126, 3085, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(54, 24, F5, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,5)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(565, 3150, F5, 15) (dual of [3150, 3085, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(565, 3144, F5, 15) (dual of [3144, 3079, 16]-code), using
(50, 65, 3219)-Net over F5 — Digital
Digital (50, 65, 3219)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(565, 3219, F5, 15) (dual of [3219, 3154, 16]-code), using
- 90 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 23 times 0, 1, 52 times 0) [i] based on linear OA(560, 3124, F5, 15) (dual of [3124, 3064, 16]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using
- 90 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 23 times 0, 1, 52 times 0) [i] based on linear OA(560, 3124, F5, 15) (dual of [3124, 3064, 16]-code), using
(50, 65, 2077019)-Net in Base 5 — Upper bound on s
There is no (50, 65, 2077020)-net in base 5, because
- 1 times m-reduction [i] would yield (50, 64, 2077020)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 542 102118 439732 050665 022317 423612 438258 359537 > 564 [i]