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Ramp secret sharing replaces the random coefficients in original Shamir secret sharing with original data to be shared. In such a case, all the coefficients in the polynomial evaluation will be secrets. If such ramp secret sharing supports additive homomorphism, how these secrets are reconstructed. Can the coefficients other than constant term can be reconstructed through Lagranage interpolation.

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It does indeed. I've written a bit about this here with code examples: https://mortendahl.github.io/2017/06/04/secret-sharing-part1/#packed-variant.

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    $\begingroup$ Welcome to Cryptography! Whilst this may theoretically answer the question, it would be preferable to include the essential parts of the answer here, and provide the link for reference. $\endgroup$
    – Andrew T.
    Jul 19, 2019 at 15:44
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    $\begingroup$ @Morten Dahl Many users of this website will be delighted to hear at least the gist of what you have written on secret sharing. $\endgroup$
    – Patriot
    Jul 19, 2019 at 16:28

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