Best Known (56, 74, s)-Nets in Base 5
(56, 74, 348)-Net over F5 — Constructive and digital
Digital (56, 74, 348)-net over F5, using
- 52 times duplication [i] based on digital (54, 72, 348)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 348, F5, 18, 18) (dual of [(348, 18), 6192, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(572, 3132, F5, 18) (dual of [3132, 3060, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(572, 3136, F5, 18) (dual of [3136, 3064, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- 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(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(51, 11, F5, 1) (dual of [11, 10, 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(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(572, 3136, F5, 18) (dual of [3136, 3064, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(572, 3132, F5, 18) (dual of [3132, 3060, 19]-code), using
- net defined by OOA [i] based on linear OOA(572, 348, F5, 18, 18) (dual of [(348, 18), 6192, 19]-NRT-code), using
(56, 74, 2617)-Net over F5 — Digital
Digital (56, 74, 2617)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(574, 2617, F5, 18) (dual of [2617, 2543, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(574, 3139, F5, 18) (dual of [3139, 3065, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- 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(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, 12, F5, 1) (dual of [12, 11, 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(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(574, 3139, F5, 18) (dual of [3139, 3065, 19]-code), using
(56, 74, 579126)-Net in Base 5 — Upper bound on s
There is no (56, 74, 579127)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 5293 993315 402505 965210 758819 957939 738711 696208 296125 > 574 [i]