This paper addresses the problem of code equivalence and its application in digital signatures. Code equivalence involves finding an isometry that maps one linear code to another. This problem has applications in cryptographic protocols, particularly in Zero-Knowledge Proof systems. The paper introduces a new notion of equivalence, which leads to a significantly reduced witness size for proving equivalence. This new approach is based on the concept of canonical representatives, which are representatives for classes of codes equivalent under some notion of equivalence. The paper proposes new notions of equivalence that extend and encompass existing ones, allowing for broader classes of equivalent codes to be identified with compact witnesses. These new notions are associated with the Canonical Form Linear Equivalence Problem (CF-LEP), which is shown to be as hard as the original code equivalence problem. The paper also presents a new solver for the code equivalence problem, which is efficient when the finite field size is large. Furthermore, the framework leads to a significant reduction in signature size compared to the LESS submission. The proposed variant can achieve very compact signatures, around 2 KB or less, which are among the smallest in the code-based setting. The paper also discusses the theoretical lower bound on the size of non-ephemeral responses in a ZK protocol based on group actions, which is 2λ. Despite recent advances, the communication cost for the code equivalence group action is still far from optimal. The paper shows that the proposed methods can significantly reduce the communication cost for proving equivalence between codes.This paper addresses the problem of code equivalence and its application in digital signatures. Code equivalence involves finding an isometry that maps one linear code to another. This problem has applications in cryptographic protocols, particularly in Zero-Knowledge Proof systems. The paper introduces a new notion of equivalence, which leads to a significantly reduced witness size for proving equivalence. This new approach is based on the concept of canonical representatives, which are representatives for classes of codes equivalent under some notion of equivalence. The paper proposes new notions of equivalence that extend and encompass existing ones, allowing for broader classes of equivalent codes to be identified with compact witnesses. These new notions are associated with the Canonical Form Linear Equivalence Problem (CF-LEP), which is shown to be as hard as the original code equivalence problem. The paper also presents a new solver for the code equivalence problem, which is efficient when the finite field size is large. Furthermore, the framework leads to a significant reduction in signature size compared to the LESS submission. The proposed variant can achieve very compact signatures, around 2 KB or less, which are among the smallest in the code-based setting. The paper also discusses the theoretical lower bound on the size of non-ephemeral responses in a ZK protocol based on group actions, which is 2λ. Despite recent advances, the communication cost for the code equivalence group action is still far from optimal. The paper shows that the proposed methods can significantly reduce the communication cost for proving equivalence between codes.