Best Known (96, 96+23, s)-Nets in Base 5
(96, 96+23, 1424)-Net over F5 — Constructive and digital
Digital (96, 119, 1424)-net over F5, using
- net defined by OOA [i] based on linear OOA(5119, 1424, F5, 23, 23) (dual of [(1424, 23), 32633, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5119, 15665, F5, 23) (dual of [15665, 15546, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(510, 40, F5, 5) (dual of [40, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(510, 52, F5, 5) (dual of [52, 42, 6]-code), using
- trace code [i] based on linear OA(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using
- extended Reed–Solomon code RSe(21,25) [i]
- the expurgated narrow-sense BCH-code C(I) with length 26 | 252−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- algebraic-geometric code AG(F,10P) with 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, Q+6P) with degQ = 2 and 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(255, 26, F25, 5) (dual of [26, 21, 6]-code or 26-arc in PG(4,25)), using
- discarding factors / shortening the dual code based on linear OA(510, 52, F5, 5) (dual of [52, 42, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(5119, 15665, F5, 23) (dual of [15665, 15546, 24]-code), using
(96, 96+23, 15666)-Net over F5 — Digital
Digital (96, 119, 15666)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5119, 15666, F5, 23) (dual of [15666, 15547, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5118, 15664, F5, 23) (dual of [15664, 15546, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(59, 39, F5, 5) (dual of [39, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(59, 44, F5, 5) (dual of [44, 35, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(5118, 15665, F5, 22) (dual of [15665, 15547, 23]-code), using Gilbert–Varšamov bound and bm = 5118 > Vbs−1(k−1) = 1052 429252 651017 206862 308889 282280 727080 048496 491463 835386 018873 391322 736752 986945 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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(5118, 15664, F5, 23) (dual of [15664, 15546, 24]-code), using
- construction X with Varšamov bound [i] based on
(96, 96+23, large)-Net in Base 5 — Upper bound on s
There is no (96, 119, large)-net in base 5, because
- 21 times m-reduction [i] would yield (96, 98, large)-net in base 5, but