I am a little confused with the relationship between various bounds for error correcting codes. Does the Plotkin bound mean that one can not achieve the Singleton bound asymptotically? That is, is there any class of error correcting $q$-ary codes with fixed $q$ for which $r+\delta$ goes to 1 as the length goes to infinity?
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2$\begingroup$ It is up to you, but usually questions with unclarified notations, definitions and references (even if they are standard in certain area) attract less attention. $\endgroup$– Fedor PetrovCommented May 2, 2023 at 7:24
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2$\begingroup$ The Wikipedia page on the Singleton bound has examples of families of codes for which it is tight. I don't know whether that answers the question or means that the question needs to be worded more tightly. $\endgroup$– Peter TaylorCommented May 2, 2023 at 8:42
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1$\begingroup$ Agree with the comments, tighten up your question with definitions if you want someone to pay more attention. $\endgroup$– kodluCommented May 2, 2023 at 14:24
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There do exist codes that achieve the Singleton bound, most famously Reed-Solomon codes.
The asymptotic version of the Plotkin bound states that $R \le 1 - \frac q{q-1}\delta + o(1)$, where $R$ is the rate of a $q$-ary code and $\delta$ is its normalized distance. The Singleton bound states that $R \le 1-\delta + o(1)$, so as $q$ tends to $\infty$ the bounds coincide.
I am guessing that your confusion is since you saw the Plotkin bound stated just for the binary ($q=2$) case. For $q=2$, the bound states $R \le 1- 2\delta + o(1)$ which is stronger than Singleton. Indeed, the Singleton bound cannot be attained for $q=2$ or in fact any constant $q$.
