Best Known (55, 72, s)-Nets in Base 5
(55, 72, 393)-Net over F5 — Constructive and digital
Digital (55, 72, 393)-net over F5, using
- 51 times duplication [i] based on digital (54, 71, 393)-net over F5, using
- net defined by OOA [i] based on linear OOA(571, 393, F5, 17, 17) (dual of [(393, 17), 6610, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(571, 3145, F5, 17) (dual of [3145, 3074, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(571, 3146, F5, 17) (dual of [3146, 3075, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- 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(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(54, 20, F5, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,5)), 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(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(571, 3146, F5, 17) (dual of [3146, 3075, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(571, 3145, F5, 17) (dual of [3145, 3074, 18]-code), using
- net defined by OOA [i] based on linear OOA(571, 393, F5, 17, 17) (dual of [(393, 17), 6610, 18]-NRT-code), using
(55, 72, 3173)-Net over F5 — Digital
Digital (55, 72, 3173)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(572, 3173, F5, 17) (dual of [3173, 3101, 18]-code), using
- 37 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 21 times 0) [i] based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 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(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(16) ⊂ Ce(15) [i] based on
- 37 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 21 times 0) [i] based on linear OA(566, 3130, F5, 17) (dual of [3130, 3064, 18]-code), using
(55, 72, 1503097)-Net in Base 5 — Upper bound on s
There is no (55, 72, 1503098)-net in base 5, because
- 1 times m-reduction [i] would yield (55, 71, 1503098)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 42 351665 678526 364629 188576 782040 995871 387749 849985 > 571 [i]