Best Known (118−20, 118, s)-Nets in Base 5
(118−20, 118, 7815)-Net over F5 — Constructive and digital
Digital (98, 118, 7815)-net over F5, using
- t-expansion [i] based on digital (97, 118, 7815)-net over F5, using
- net defined by OOA [i] based on linear OOA(5118, 7815, F5, 21, 21) (dual of [(7815, 21), 163997, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5118, 78151, F5, 21) (dual of [78151, 78033, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5118, 78152, F5, 21) (dual of [78152, 78034, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(5113, 78125, F5, 21) (dual of [78125, 78012, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-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(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5118, 78152, F5, 21) (dual of [78152, 78034, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5118, 78151, F5, 21) (dual of [78151, 78033, 22]-code), using
- net defined by OOA [i] based on linear OOA(5118, 7815, F5, 21, 21) (dual of [(7815, 21), 163997, 22]-NRT-code), using
(118−20, 118, 65962)-Net over F5 — Digital
Digital (98, 118, 65962)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5118, 65962, F5, 20) (dual of [65962, 65844, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(5118, 78158, F5, 20) (dual of [78158, 78040, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(5113, 78125, F5, 21) (dual of [78125, 78012, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(55, 33, F5, 3) (dual of [33, 28, 4]-code or 33-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
- discarding factors / shortening the dual code based on linear OA(5118, 78158, F5, 20) (dual of [78158, 78040, 21]-code), using
(118−20, 118, large)-Net in Base 5 — Upper bound on s
There is no (98, 118, large)-net in base 5, because
- 18 times m-reduction [i] would yield (98, 100, large)-net in base 5, but