Best Known (135−28, 135, s)-Nets in Base 5
(135−28, 135, 1117)-Net over F5 — Constructive and digital
Digital (107, 135, 1117)-net over F5, using
- 51 times duplication [i] based on digital (106, 134, 1117)-net over F5, using
- net defined by OOA [i] based on linear OOA(5134, 1117, F5, 28, 28) (dual of [(1117, 28), 31142, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(5134, 15638, F5, 28) (dual of [15638, 15504, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(5133, 15625, F5, 28) (dual of [15625, 15492, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(5121, 15625, F5, 26) (dual of [15625, 15504, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(27) ⊂ Ce(25) [i] based on
- OA 14-folding and stacking [i] based on linear OA(5134, 15638, F5, 28) (dual of [15638, 15504, 29]-code), using
- net defined by OOA [i] based on linear OOA(5134, 1117, F5, 28, 28) (dual of [(1117, 28), 31142, 29]-NRT-code), using
(135−28, 135, 10540)-Net over F5 — Digital
Digital (107, 135, 10540)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5135, 10540, F5, 28) (dual of [10540, 10405, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5135, 15639, F5, 28) (dual of [15639, 15504, 29]-code), using
- 1 times code embedding in larger space [i] based on linear OA(5134, 15638, F5, 28) (dual of [15638, 15504, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(5133, 15625, F5, 28) (dual of [15625, 15492, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(5121, 15625, F5, 26) (dual of [15625, 15504, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(27) ⊂ Ce(25) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(5134, 15638, F5, 28) (dual of [15638, 15504, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5135, 15639, F5, 28) (dual of [15639, 15504, 29]-code), using
(135−28, 135, 8307400)-Net in Base 5 — Upper bound on s
There is no (107, 135, 8307401)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 22958 911176 068118 119897 793098 641832 631072 643598 565657 687198 269270 958842 397378 683448 633592 322025 > 5135 [i]