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<p>Let <i>F</i> be any field containing the finite field of order <i>q</i>. A <i>q</i>-polynomial <i>L</i> over <i>F</i> is an element of the polynomial ring <i>F</i>[<i>x</i>] with the property that all powers of <i>x</i> that appear in <i>L</i> with nonzero coefficient have exponent a power of <i>q</i>. It is well known that given any ordinary polynomial <i>f</i> in <i>F</i>[<i>x</i>], there exists a <i>q</i>-polynomial that is divisible by <i>f</i>. We study the smallest degree of such a <i>q</i>-polynomial. This is equivalent to studying the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{\,\mathrm{\mathbb {F}}\,}}_q\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mspace width="0.166667em" /> <mi mathvariant="double-struck">F</mi> <mspace width="0.166667em" /> </mrow> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-span of the roots of <i>f</i> in a splitting field. We relate this quantity to the representation theory of the Galois group of <i>f</i>. As an application we give a simultaneous construction of the binary Golay code of length 24, and the Steiner system on 24 points.</p>

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Linearization of polynomials in prime characteristic, with applications to the Golay code and Steiner system

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  • 发表日期：

    2025年8月27日

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  • Rod Gow,
  • Gary McGuire

摘要

Let F be any field containing the finite field of order q. A q-polynomial L over F is an element of the polynomial ring F[x] with the property that all powers of x that appear in L with nonzero coefficient have exponent a power of q. It is well known that given any ordinary polynomial f in F[x], there exists a q-polynomial that is divisible by f. We study the smallest degree of such a q-polynomial. This is equivalent to studying the \({{\,\mathrm{\mathbb {F}}\,}}_q\) F q -span of the roots of f in a splitting field. We relate this quantity to the representation theory of the Galois group of f. As an application we give a simultaneous construction of the binary Golay code of length 24, and the Steiner system on 24 points.