Best Known (121−20, 121, s)-Nets in Base 5
(121−20, 121, 7816)-Net over F5 — Constructive and digital
Digital (101, 121, 7816)-net over F5, using
- t-expansion [i] based on digital (100, 121, 7816)-net over F5, using
- net defined by OOA [i] based on linear OOA(5121, 7816, F5, 21, 21) (dual of [(7816, 21), 164015, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5121, 78161, F5, 21) (dual of [78161, 78040, 22]-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(58, 36, F5, 4) (dual of [36, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(58, 52, F5, 4) (dual of [52, 44, 5]-code), using
- trace code [i] based on linear OA(254, 26, F25, 4) (dual of [26, 22, 5]-code or 26-arc in PG(3,25)), using
- extended Reed–Solomon code RSe(22,25) [i]
- algebraic-geometric code AG(F, Q+9P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using the rational function field F25(x) [i]
- algebraic-geometric code AG(F,7P) with degPÂ =Â 3 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26 (see above)
- trace code [i] based on linear OA(254, 26, F25, 4) (dual of [26, 22, 5]-code or 26-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(58, 52, F5, 4) (dual of [52, 44, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(5121, 78161, F5, 21) (dual of [78161, 78040, 22]-code), using
- net defined by OOA [i] based on linear OOA(5121, 7816, F5, 21, 21) (dual of [(7816, 21), 164015, 22]-NRT-code), using
(121−20, 121, 78163)-Net over F5 — Digital
Digital (101, 121, 78163)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5121, 78163, F5, 20) (dual of [78163, 78042, 21]-code), using
- construction X with Varšamov bound [i] 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
- linear OA(5118, 78160, F5, 18) (dual of [78160, 78042, 19]-code), using Gilbert–Varšamov bound and bm = 5118 > Vbs−1(k−1) = 7 309069 901290 073384 883047 833310 024926 645612 110152 243656 144992 535681 611005 159565 [i]
- linear OA(51, 3, F5, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(5118, 78158, F5, 20) (dual of [78158, 78040, 21]-code), using
- construction X with Varšamov bound [i] based on
(121−20, 121, large)-Net in Base 5 — Upper bound on s
There is no (101, 121, large)-net in base 5, because
- 18 times m-reduction [i] would yield (101, 103, large)-net in base 5, but