Semiconductor ring lasers (SRLs) naturally support clockwise (CW) and counterclockwise (CCW) counter-propagating modes, and thus offer intrinsic dual-channel parallelism in a compact cavity well suited to photonic integration. However, photonic reservoir computing (RC) built on such lasers usually relies on an external optical feedback loop to provide fading memory, which makes the operating parameters difficult to calibrate and confines the system to a narrow stable operating range. In this work, a feedback-free photonic RC scheme is proposed by combining a single SRL biased in its bidirectional emission regime with bio-inspired quasi-convolution (QC) coding, in which a triangular kernel with linearly decreasing weights algorithmically embeds tunable short-term memory into the data stream. The kernel size Q and the step coefficient \beta independently control the depth and the temporal granularity of this memory. According to where the kernel is inserted, three complementary variants are constructed, an input-coding variant (IC-RC), in which the CW mode is driven by the uncoded masked signal and the CCW mode by the coded signal, so that the two modes acquire differentiated temporal contexts, an output-coding variant (OC-RC), in which the kernel instead acts on the detected mode intensities, and a hybrid-coding variant (HC-RC), in which independent kernels are applied at both the input and the output. The two mode intensities are sampled and concatenated into a single readout matrix, which doubles the number of virtual nodes without additional hardware. Based on a rate-equation model, the three schemes are benchmarked against a conventional time-delay reservoir computer (TDRC) on Santa Fe chaotic time-series prediction, nonlinear channel equalization, and linear memory capacity (MC). All three variants outperform the TDRC and maintain low, plateau-like errors over the entire injection-strength scan, broadening the high-performance region in the plane of injection strength and frequency detuning by more than an order of magnitude. The two families of tasks are found to be governed by different kernel properties, prediction and equalization depend on the overall richness of the temporal context and therefore vary smoothly with the kernel parameters, whereas the memory capacity depends on the precise alignment between the kernel delay and the sampling grid and thus rises in discrete steps. IC-RC achieves the highest prediction accuracy and HC-RC the largest memory capacity, while the output-layer activation function further improves prediction and equalization but has little effect on the MC. These results provide a theoretical basis for feedback-free photonic RC and offer new insight into the development of highly integrable photonic neuromorphic computing systems.