MinT - Best Known (123, 123+22, s)-Nets in Base 5

Best Known (123, 123+22, s)-Nets in Base 5

(123, 123+22, 35515)-Net over F₅ — Constructive and digital

Digital (123, 145, 35515)-net over F₅, using

net defined by OOA [i] based on linear OOA(5¹⁴⁵, 35515, F₅, 22, 22) (dual of [(35515, 22), 781185, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5¹⁴⁵, 390665, F₅, 22) (dual of [390665, 390520, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(16) [i] based on
    1. linear OA(5¹³⁷, 390625, F₅, 22) (dual of [390625, 390488, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(5¹⁰⁵, 390625, F₅, 17) (dual of [390625, 390520, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(5⁸, 40, F₅, 4) (dual of [40, 32, 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 Reedâ€“Solomon code RS_e(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)

(123, 123+22, 223771)-Net over F₅ — Digital

Digital (123, 145, 223771)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴⁵, 223771, F₅, 22) (dual of [223771, 223626, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁴⁵, 390665, F₅, 22) (dual of [390665, 390520, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(16) [i] based on
    1. linear OA(5¹³⁷, 390625, F₅, 22) (dual of [390625, 390488, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(5¹⁰⁵, 390625, F₅, 17) (dual of [390625, 390520, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(5⁸, 40, F₅, 4) (dual of [40, 32, 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 Reedâ€“Solomon code RS_e(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)

(123, 123+22, large)-Net in Base 5 — Upper bound on s

There is no (123, 145, large)-net in base 5, because

20 times m-reduction [i] would yield (123, 125, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (123, 123+22, s)-Nets in Base 5

(123, 123+22, 35515)-Net over F5 — Constructive and digital

(123, 123+22, 223771)-Net over F5 — Digital

(123, 123+22, large)-Net in Base 5 — Upper bound on s

(123, 123+22, 35515)-Net over F₅ — Constructive and digital

(123, 123+22, 223771)-Net over F₅ — Digital