Best Known (55−13, 55, s)-Nets in Base 5
(55−13, 55, 523)-Net over F5 — Constructive and digital
Digital (42, 55, 523)-net over F5, using
- 51 times duplication [i] based on digital (41, 54, 523)-net over F5, using
- net defined by OOA [i] based on linear OOA(554, 523, F5, 13, 13) (dual of [(523, 13), 6745, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(554, 3139, F5, 13) (dual of [3139, 3085, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(552, 3137, F5, 13) (dual of [3137, 3085, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(551, 3126, F5, 13) (dual of [3126, 3075, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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(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 C([0,6]) ⊂ C([0,5]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(552, 3137, F5, 13) (dual of [3137, 3085, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(554, 3139, F5, 13) (dual of [3139, 3085, 14]-code), using
- net defined by OOA [i] based on linear OOA(554, 523, F5, 13, 13) (dual of [(523, 13), 6745, 14]-NRT-code), using
(55−13, 55, 3164)-Net over F5 — Digital
Digital (42, 55, 3164)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(555, 3164, F5, 13) (dual of [3164, 3109, 14]-code), using
- 30 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0) [i] based on linear OA(551, 3130, F5, 13) (dual of [3130, 3079, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- 30 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0) [i] based on linear OA(551, 3130, F5, 13) (dual of [3130, 3079, 14]-code), using
(55−13, 55, 1461810)-Net in Base 5 — Upper bound on s
There is no (42, 55, 1461811)-net in base 5, because
- 1 times m-reduction [i] would yield (42, 54, 1461811)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 55 511368 670817 268527 025284 000984 110825 > 554 [i]