Best Known (53, 53+13, s)-Nets in Base 5
(53, 53+13, 2608)-Net over F5 — Constructive and digital
Digital (53, 66, 2608)-net over F5, using
- net defined by OOA [i] based on linear OOA(566, 2608, F5, 13, 13) (dual of [(2608, 13), 33838, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(566, 15649, F5, 13) (dual of [15649, 15583, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- linear OA(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(543, 15625, F5, 9) (dual of [15625, 15582, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(54, 23, F5, 3) (dual of [23, 19, 4]-code or 23-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(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(566, 15649, F5, 13) (dual of [15649, 15583, 14]-code), using
(53, 53+13, 15649)-Net over F5 — Digital
Digital (53, 66, 15649)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(566, 15649, F5, 13) (dual of [15649, 15583, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- linear OA(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(543, 15625, F5, 9) (dual of [15625, 15582, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(54, 23, F5, 3) (dual of [23, 19, 4]-code or 23-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(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
(53, 53+13, large)-Net in Base 5 — Upper bound on s
There is no (53, 66, large)-net in base 5, because
- 11 times m-reduction [i] would yield (53, 55, large)-net in base 5, but