<p>Quasi-2-dimensional cyclic (Q2DC) codes are a special type of asymptotically good linear codes. Based on the results on Gauss sums and cyclotomic field <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3386_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q(\zeta _{p})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <msub> <mi>ζ</mi> <mi>p</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we mainly construct some classes of Q2DC codes and determine their complete weight enumerators. As applications, we derive a class of constant composition codes which achieve the LFVC bound. Furthermore, we investigate the minimality of our codes, some new families of minimal Q2DC codes with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3386_Article_IEq2.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{w_{min}}{w_{max}}\le \frac{p-1}{p}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <msub> <mi>w</mi> <mrow> <mi mathvariant="italic">min</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi mathvariant="italic">max</mi> </mrow> </msub> </mfrac> <mo>≤</mo> <mfrac> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>p</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> are constructed.</p>