MinT - Best Known (142−24, 142, s)-Nets in Base 5

Best Known (142−24, 142, s)-Nets in Base 5

(142−24, 142, 6513)-Net over F₅ — Constructive and digital

Digital (118, 142, 6513)-net over F₅, using

5¹ times duplication [i] based on digital (117, 141, 6513)-net over F₅, using
- net defined by OOA [i] based on linear OOA(5¹⁴¹, 6513, F₅, 24, 24) (dual of [(6513, 24), 156171, 25]-NRT-code), using
  - OA 12-folding and stacking [i] based on linear OA(5¹⁴¹, 78156, F₅, 24) (dual of [78156, 78015, 25]-code), using
    - constructionÂ XX applied to C_e(23) âŠ‚ C_e(20) âŠ‚ C_e(18) [i] based on
      1. linear OA(5¹³⁴, 78125, F₅, 24) (dual of [78125, 77991, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
      2. linear OA(5¹¹³, 78125, F₅, 21) (dual of [78125, 78012, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      3. linear OA(5¹⁰⁶, 78125, F₅, 19) (dual of [78125, 78019, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
      4. linear OA(5³, 27, F₅, 2) (dual of [27, 24, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(5³, 31, F₅, 2) (dual of [31, 28, 3]-code), using
        Hamming code H(3,5) [i]
      5. linear OA(5¹, 4, F₅, 1) (dual of [4, 3, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(5¹, 5, F₅, 1) (dual of [5, 4, 2]-code), using
        Reedâ€“Solomon code RS(4,5) [i]

(142−24, 142, 68306)-Net over F₅ — Digital

Digital (118, 142, 68306)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴², 68306, F₅, 24) (dual of [68306, 68164, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁴², 78161, F₅, 24) (dual of [78161, 78019, 25]-code), using
  - constructionÂ X applied to C_e(23) âŠ‚ C_e(18) [i] based on
    1. linear OA(5¹³⁴, 78125, F₅, 24) (dual of [78125, 77991, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
    2. linear OA(5¹⁰⁶, 78125, F₅, 19) (dual of [78125, 78019, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. 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 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)

(142−24, 142, large)-Net in Base 5 — Upper bound on s

There is no (118, 142, large)-net in base 5, because

22 times m-reduction [i] would yield (118, 120, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (142−24, 142, s)-Nets in Base 5

(142−24, 142, 6513)-Net over F5 — Constructive and digital

(142−24, 142, 68306)-Net over F5 — Digital

(142−24, 142, large)-Net in Base 5 — Upper bound on s

(142−24, 142, 6513)-Net over F₅ — Constructive and digital

(142−24, 142, 68306)-Net over F₅ — Digital