Best Known (78−19, 78, s)-Nets in Base 5
(78−19, 78, 348)-Net over F5 — Constructive and digital
Digital (59, 78, 348)-net over F5, using
- 51 times duplication [i] based on digital (58, 77, 348)-net over F5, using
- net defined by OOA [i] based on linear OOA(577, 348, F5, 19, 19) (dual of [(348, 19), 6535, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(577, 3133, F5, 19) (dual of [3133, 3056, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(577, 3136, F5, 19) (dual of [3136, 3059, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- 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(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(577, 3136, F5, 19) (dual of [3136, 3059, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(577, 3133, F5, 19) (dual of [3133, 3056, 20]-code), using
- net defined by OOA [i] based on linear OOA(577, 348, F5, 19, 19) (dual of [(348, 19), 6535, 20]-NRT-code), using
(78−19, 78, 2618)-Net over F5 — Digital
Digital (59, 78, 2618)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(578, 2618, F5, 19) (dual of [2618, 2540, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(578, 3138, F5, 19) (dual of [3138, 3060, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- 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(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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, 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(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(18) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(578, 3138, F5, 19) (dual of [3138, 3060, 20]-code), using
(78−19, 78, 990296)-Net in Base 5 — Upper bound on s
There is no (59, 78, 990297)-net in base 5, because
- 1 times m-reduction [i] would yield (59, 77, 990297)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 661745 090561 362897 692071 954759 964452 928658 341268 853445 > 577 [i]