In 2012, Press and Dyson discovered a strategy set, called Zero-determinant (ZD) strategies, which enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy in the repeated prisoner's dilemma (RPD) game. In the RPD game, a discount factor and observation errors are both important because they often happen in society. Here, we examined strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first revealed that the payoffs of two players can be represented by the form of determinants even with these two factors. Then, we searched for all possible strategies that enforce linear payoff relationships and found that both ZD strategies and unconditional strategies are the only strategy sets which satisfy the condition.