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We study (1 + 2<i>v</i>)-constacyclic codes over<i>R</i> + <i>vR</i> and their Gray images, where <i>v</i>² + <i>v</i> = 0 and R is a finite chain ring with maximal ideal <<i>λ</i>> and nilpotency index <i>e.</i> It is proved that the Gray map images of a (1 + 2<i>v</i>)-constacyclic codes of length <i>n</i> over <i>R</i> + <i>vR</i> are distance-invariant linear cyclic codes of length 2<i>n</i> over <i>R</i>. The generator polynomials of this kind of codes for length <i>n</i> are determined, where n is relatively prime to <i>p</i>, <i>p</i> is the character of the field <i>R</i>/<<i>λ</i>> . Their dual codes are also discussed

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The matrix description of a near-MDR code is given, and some judging criterions are presented for near-MDR codes. We also give the weight distribution of a near-MDR code and the applications of a near-MDR code to secret sharing schemes. Furthermore, we will introduce the chain condition for free codes over finite chain rings, and then present a formula for computing higher weights of tensor product of free codes satisfying the chain condition. We will also find a chain for any near-MDR code, and thus show that any near-MDR code satisfies the chain condition.