MinT - Best Known (111−22, 111, s)-Nets in Base 5

Best Known (111−22, 111, s)-Nets in Base 5

(111−22, 111, 1423)-Net over F₅ — Constructive and digital

Digital (89, 111, 1423)-net over F₅, using

5² times duplication [i] based on digital (87, 109, 1423)-net over F₅, using
- net defined by OOA [i] based on linear OOA(5¹⁰⁹, 1423, F₅, 22, 22) (dual of [(1423, 22), 31197, 23]-NRT-code), using
  - OA 11-folding and stacking [i] based on linear OA(5¹⁰⁹, 15653, F₅, 22) (dual of [15653, 15544, 23]-code), using
    - discarding factors / shortening the dual code based on linear OA(5¹⁰⁹, 15655, F₅, 22) (dual of [15655, 15546, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(16) [i] based on
        linear OA(5¹⁰³, 15625, F₅, 22) (dual of [15625, 15522, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(5⁷⁹, 15625, F₅, 17) (dual of [15625, 15546, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
        linear OA(5⁶, 30, F₅, 4) (dual of [30, 24, 5]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(111−22, 111, 14493)-Net over F₅ — Digital

Digital (89, 111, 14493)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹¹¹, 14493, F₅, 22) (dual of [14493, 14382, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹¹¹, 15657, F₅, 22) (dual of [15657, 15546, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(16) [i] based on
    1. linear OA(5¹⁰³, 15625, F₅, 22) (dual of [15625, 15522, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(5⁷⁹, 15625, F₅, 17) (dual of [15625, 15546, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(5⁸, 32, F₅, 4) (dual of [32, 24, 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)

(111−22, 111, large)-Net in Base 5 — Upper bound on s

There is no (89, 111, large)-net in base 5, because

20 times m-reduction [i] would yield (89, 91, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (111−22, 111, s)-Nets in Base 5

(111−22, 111, 1423)-Net over F5 — Constructive and digital

(111−22, 111, 14493)-Net over F5 — Digital

(111−22, 111, large)-Net in Base 5 — Upper bound on s

(111−22, 111, 1423)-Net over F₅ — Constructive and digital

(111−22, 111, 14493)-Net over F₅ — Digital