Best Known (84, 84+20, s)-Nets in Base 5
(84, 84+20, 1565)-Net over F5 — Constructive and digital
Digital (84, 104, 1565)-net over F5, using
- 51 times duplication [i] based on digital (83, 103, 1565)-net over F5, using
- t-expansion [i] based on digital (82, 103, 1565)-net over F5, using
- net defined by OOA [i] based on linear OOA(5103, 1565, F5, 21, 21) (dual of [(1565, 21), 32762, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5103, 15651, F5, 21) (dual of [15651, 15548, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(5103, 15651, F5, 21) (dual of [15651, 15548, 22]-code), using
- net defined by OOA [i] based on linear OOA(5103, 1565, F5, 21, 21) (dual of [(1565, 21), 32762, 22]-NRT-code), using
- t-expansion [i] based on digital (82, 103, 1565)-net over F5, using
(84, 84+20, 15658)-Net over F5 — Digital
Digital (84, 104, 15658)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5104, 15658, F5, 20) (dual of [15658, 15554, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5102, 15654, F5, 20) (dual of [15654, 15552, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(55, 29, F5, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(5102, 15656, F5, 19) (dual of [15656, 15554, 20]-code), using Gilbert–Varšamov bound and bm = 5102 > Vbs−1(k−1) = 33913 535144 688648 156537 961687 293412 555337 358822 851281 556968 524059 797165 [i]
- 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
- linear OA(5102, 15654, F5, 20) (dual of [15654, 15552, 21]-code), using
- construction X with Varšamov bound [i] based on
(84, 84+20, large)-Net in Base 5 — Upper bound on s
There is no (84, 104, large)-net in base 5, because
- 18 times m-reduction [i] would yield (84, 86, large)-net in base 5, but