Best Known (70, 88, s)-Nets in Base 5
(70, 88, 1737)-Net over F5 — Constructive and digital
Digital (70, 88, 1737)-net over F5, using
- 52 times duplication [i] based on digital (68, 86, 1737)-net over F5, using
- net defined by OOA [i] based on linear OOA(586, 1737, F5, 18, 18) (dual of [(1737, 18), 31180, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(586, 15633, F5, 18) (dual of [15633, 15547, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(586, 15638, F5, 18) (dual of [15638, 15552, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(585, 15625, F5, 18) (dual of [15625, 15540, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(51, 13, F5, 1) (dual of [13, 12, 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(586, 15638, F5, 18) (dual of [15638, 15552, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(586, 15633, F5, 18) (dual of [15633, 15547, 19]-code), using
- net defined by OOA [i] based on linear OOA(586, 1737, F5, 18, 18) (dual of [(1737, 18), 31180, 19]-NRT-code), using
(70, 88, 10732)-Net over F5 — Digital
Digital (70, 88, 10732)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(588, 10732, F5, 18) (dual of [10732, 10644, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(588, 15641, F5, 18) (dual of [15641, 15553, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(585, 15625, F5, 18) (dual of [15625, 15540, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 14, F5, 1) (dual of [14, 13, 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(588, 15641, F5, 18) (dual of [15641, 15553, 19]-code), using
(70, 88, 7080508)-Net in Base 5 — Upper bound on s
There is no (70, 88, 7080509)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 32 311762 769454 054991 953669 833113 031481 461001 350077 408296 041045 > 588 [i]