Best Known (87, 87+22, s)-Nets in Base 5
(87, 87+22, 1423)-Net over F5 — Constructive and digital
Digital (87, 109, 1423)-net over F5, using
- net defined by OOA [i] based on linear OOA(5109, 1423, F5, 22, 22) (dual of [(1423, 22), 31197, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5109, 15653, F5, 22) (dual of [15653, 15544, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5109, 15655, F5, 22) (dual of [15655, 15546, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(5103, 15625, F5, 22) (dual of [15625, 15522, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5109, 15655, F5, 22) (dual of [15655, 15546, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(5109, 15653, F5, 22) (dual of [15653, 15544, 23]-code), using
(87, 87+22, 12337)-Net over F5 — Digital
Digital (87, 109, 12337)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5109, 12337, F5, 22) (dual of [12337, 12228, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5109, 15651, F5, 22) (dual of [15651, 15542, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- linear OA(5103, 15625, F5, 22) (dual of [15625, 15522, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(585, 15625, F5, 18) (dual of [15625, 15540, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(54, 24, F5, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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 XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5109, 15651, F5, 22) (dual of [15651, 15542, 23]-code), using
(87, 87+22, large)-Net in Base 5 — Upper bound on s
There is no (87, 109, large)-net in base 5, because
- 20 times m-reduction [i] would yield (87, 89, large)-net in base 5, but