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    "title": "Goppa: binary error-correcting codes beyond cyclic constructions",
    "summary": "Goppa's1970 paper defines binary linear codes through rational functions modulo a polynomial, giving dimension bounds and a stronger designed-distance guarantee when that polynomial has distinct roots.\n\nRedundant bits can help recover a damaged message. Goppa constructs a family where the field and chosen polynomial control the redundancy and correction guarantee without generally requiring cyclic symmetry. For square-free degree-t polynomials in the stated binary setting, the minimum distance is at least2t+1, so up to t changed bits can be uniquely corrected in principle.",
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        "description": "The exact target is the binary finite-field rational-function code construction of Sections 2–4, with the conditional distance bound and the paper's stated decoding outline. Support L consists of distinct elements of GF(2^m), excluding roots of the defining polynomial; dimension is at least n-mt when its degree is t. The 2 t+1 distance bound retained here requires distinct polynomial roots, as Section 4 specifies; it is not asserted for arbitrary repeated-root polynomials. Minimum-distance guarantees are algebraic error limits, not performance under every communication channel. The source describes a decoding scheme and example. This account does not supply a verified implementation, speed comparison, independently replayed example or exact decoder transcription. Section 8 explicitly limits this paper to binary codes; later nonbinary, algebraic-geometric and cryptographic constructions/security claims remain separate. The cyclic-code characterization has additional Section 7 support/group conditions and is not newly asserted universally. Primary text was inspected through a public web reader, not a visually checked locally authenticated PDF or independent proof audit.",
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        {
          "citation": "V.D.Goppa, A New Class of Linear Correcting Codes, Probl.Peredachi Inf.6(3)(1970),24–30, original Russian PDF text",
          "url": "https://www.mathnet.ru/php/getFT.phtml?jrnid=ppi&option_lang=eng&paperid=1748&what=fullt"
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      "statement": "# Goppa: binary error-correcting codes beyond cyclic constructions\n\nGoppa's1970 paper defines binary linear codes through rational functions modulo a polynomial, giving dimension bounds and a stronger designed-distance guarantee when that polynomial has distinct roots.\n\nRedundant bits can help recover a damaged message. Goppa constructs a family where the field and chosen polynomial control the redundancy and correction guarantee without generally requiring cyclic symmetry. For square-free degree-t polynomials in the stated binary setting, the minimum distance is at least2t+1, so up to t changed bits can be uniquely corrected in principle.\n\n## Scientific scope\n\nThe exact target is the binary finite-field rational-function code construction of Sections 2–4, with the conditional distance bound and the paper's stated decoding outline. Support L consists of distinct elements of GF(2^m), excluding roots of the defining polynomial; dimension is at least n-mt when its degree is t. The 2 t+1 distance bound retained here requires distinct polynomial roots, as Section 4 specifies; it is not asserted for arbitrary repeated-root polynomials. Minimum-distance guarantees are algebraic error limits, not performance under every communication channel. The source describes a decoding scheme and example. This account does not supply a verified implementation, speed comparison, independently replayed example or exact decoder transcription. Section 8 explicitly limits this paper to binary codes; later nonbinary, algebraic-geometric and cryptographic constructions/security claims remain separate. The cyclic-code characterization has additional Section 7 support/group conditions and is not newly asserted universally. Primary text was inspected through a public web reader, not a visually checked locally authenticated PDF or independent proof audit.\n\n## People and systems\n\nValery D. Goppa\n\n\n\n## Original sources\n\n- [V.D.Goppa, A New Class of Linear Correcting Codes, Probl.Peredachi Inf.6(3)(1970),24–30, original Russian PDF text](https://www.mathnet.ru/php/getFT.phtml?jrnid=ppi&option_lang=eng&paperid=1748&what=fullt)\n- [Math-Net original Goppa1970 paper bibliography and English abstract](https://www.mathnet.ru/eng/ppi1748)\n",
      "summary": "Goppa's1970 paper defines binary linear codes through rational functions modulo a polynomial, giving dimension bounds and a stronger designed-distance guarantee when that polynomial has distinct roots.\n\nRedundant bits can help recover a damaged message. Goppa constructs a family where the field and chosen polynomial control the redundancy and correction guarantee without generally requiring cyclic symmetry. For square-free degree-t polynomials in the stated binary setting, the minimum distance is at least2t+1, so up to t changed bits can be uniquely corrected in principle.",
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        "title": "V. D. Goppa, A New Class of Linear Correcting Codes, Probl. Peredachi Inf.6(3)(1970),24–30, original Russian PDF text",
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